An Introduction to Nonlinear Boundary Value Problems (Mathematics in Science and Engineering, V. 109)

An Introduction to Nonlinear Boundary Value Problems This is Volume 109 in MATHEMATICS IN SCIENCE AND ENGINEERING A s...

Author: Stephen R. Bernfeld | V. Lakshmikantham

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An Introduction to Nonlinear Boundary Value Problems

This is Volume 109 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

An Introduction to Nonlinear BoundaryValue Problems Stephen R. Bernfeld Department of Mathematics Memphis State University Memphis, Tennessee

V Lakshmtkantham Department of Mathematics University of Texas Arlington, Texas

Academic Press, Inc.

New York and London

I974

A Subsidiary of Harcourt Brace Jovanovich. Publishers

COPYRIGHT 0 1974, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenw, New York. New York loo03

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON)LTD. 24/28 Oval Road, London NWl

Library of Congress Cataloging in Publication Data Bernfeld, Stephen R An introduction to nonlinear boundary value problems. (Mathematics in science and engineering, v. ) Bibliography: p. 1. Boundary value problems. 2. Nonlinear theories. I. Lakshmikantham,V., joint author. 11. Title. 111. Series. QA319.B41 515l.35 73-21996

ISBN 0-12-093150-8

Contents Preface . . . Acknowledgments

ChuprerI 1 .0. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. 1.12. 1.13. 1.14. 1.15. Chapter 2

2.0. 2.1. 2.2. 2.3.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MethodsInvolvingDifferentialInequalities . Introduction . . . . . . . . Existencein thesmall . . . . . Upper and Lower Solutions . . . TheModifiedFunction . . . . . Nagumo’s Condition . . . . . Existence in the Large . . . . . Lyapunov-Like Functions . . . Existence on Infinite Intervals . . Super-and Subfunctions . . . . Properties of Subfunctions . . . Perron’s Method . . . . . . . ModifiedVectorFunction . . . Nagumo’s Condition (Continued) . Existence in the Large for Systems. Further Results for Systems . . . Notesandcomments . . . . .

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ix

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. . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . Shooting Type Methods

Uniqueness Implies Existence . . . General Linear Boundary Conditions . Weaker Uniqueness Conditions . . . V

1 1 2 12 18 25 31 39 44 46 52 62 69 74 81 84 93 94 94

. . . . . . . . 94 . . . . . . . . 101 . . . . . . . . 109

CONTENTS

2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.1 1. 2.12.

Chapter 3 3.0. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.

Chapter 4 4.0. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.1 1. 4.12. 4.13. 4.14. 4.15. 4.16. 4.17.

Nonlinear Boundary Conditions . . . Angular Function Technique. . . . . Fundamental Lemmas . . . . . . . Existence . . . . . . . . . . . Uniqueness . . . . . . . . . . Estimation of Number of Solutions . . Existence of Infinite Number of Solutions Nonlinear Boundary Conditions . . . Notes and Comments . . . . . . . Topological Methods

. . . . . . . 113 . . . . . . . 116

. . . . . . . 117 . . . . . . . 121 . . . . . . . 127

. . . . . . . 136

. . . . . . . 142 . . . . . . . 145 . . . . . . . 152

. . . . . . . . . . . . . .

Introduction . . . . . . . . . . . Solution Funnels . . . . . . . . . Application to Second-Order Equations . . Wazewski Retract Method . . . . . . Generalized Differential Equations . . . Dependence ofSolutions on Boundary Data Notes and Comments . . . . . . . . Functional Analytic Methods

. . . . . . 153 . . . . . . 153

. . . . . . 160 . . . . . 168 . . . . . 175

. . . .

. . . . . 186 . . . . . 195

. . . . . . . . . . . 197

Introduction . . . . . . . . . . . . Linear Problems for Linear Systems . . . . Linear Problems for Nonlinear Systems . . . Interpolation Problems . . . . . . . . Further Nonlinear Problems . . . . . . . Generalized Spaces . . . . . . . . . . Integral Equations . . . . . . . . . . Application to Existence and Uniqueness . . Method of A Priori Estimates . . . . . . Bounds for Solutions in Admissible Subspaces . b r a y Schauder’s Alternative . . . . . . Application of LeraySchauder’s Alternative . Periodic Boundary Conditions . . . . . . Set-Valued Mappings and Functional Equations General Linear Problems . . . . . . . . General Results for Set-Valued Mappings . . Set-Valued Differential Equations . . . . . Notes and Comments . . . . . . . . . vi

153

. . . . . 197

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. . . . 198 . . . . 205

. . . . 208 . . . . 213 . . . . 225

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . 228

. . . . . . . . . . .

. . . . . . . . . . .

. 232 . 248 . 256 . 263 . 265 . 269 . 278 . 282 . 289 . 295 . 302

CONTENTS

. . . . . 304 Introduction . . . . . . . . . . . . . . . . . 304 Existence in the Small . . . . . . . . . . . . . . 304

Chapter5

Extensions to Functional Differential Equations

5.0. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

Existence in the Large . . . . . . . . . Shooting Method . . . . . . . . . . Nonhomogeneous Linear Boundary Conditions Linear Problems . . . . . . . . . . . Nonlinear Problems. . . . . . . . . . Degenerate Cases . . . . . . . . . . Notes and Comments . . . . . . . . .

Chapter 6 6.0. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6.

. . . . . .

. . . . . .

. . . . . .

. 312 . 315 . 320 . 324 . 330 . 335

. . . . . . . . . . . . . . . . 337 Introduction . . . . . . . . . . . . . . . . . 337 Newton's Method . . . . . . . . . . . . . . . 337

Selected Topics

The Goodman-Lance Method . . . . . . . The Method of Quasilinearization . . . . . . Nonlinear Eigenvalue Problems . . . . . . . n-Parameter Families and Interpolation Problems . Notes and Comments . . . . . . . . . .

Bibliography . . . . Additional Bibliography Index

. . . . . 308

. . . . . .

. . . . 344 . . . . 349 . . . . 353

. . . . 358

. . . . 367

. . . . . . . . . . . . . . . 368 . . . . . . . . . . . . . . . 382

. . . . . . . . . . . . . . . . . . . . . .

vii

385

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'Ihe theory of nonlinear boundary value problems is an extremely important and interesting area of research in differential equations. Due to the entirely different nature of the underlying physical processes, its study is substantially more difficult than that of initial value problems and consequently belongs to a third course in differential equations. Although this sophisticated branch of research has, in recent years, developed significantly, the available books are either more elementary in nature, for example the book by Baily, Shampine, and Waltman, or directed to a particular method of importance, such as that by Bellman and Kalaba. Hence it is felt that a book on an advanced level that exposes the reader to this fascinating field of differential equations and provides a ready access to an up-to-date state of this art is of immense value. With this as motivation, we present in our book a variety of techniques that are employed in the theory of nonlinear boundary value problems. For example, we discuss the following: (i) methods that involve differentid inequalities; (ii) shooting and angular function techniques; (iii) functional analytic approaches; (iv) topological methods. We have also included a chapter on nonlinear boundary value problems for functional differential equations and a chapter covering special topics of interest. The main features of the book are (i) a coverage of a portion of the material from the contribution of Russian mathematics of which the English speaking world is not well aware; (ii) the use of several Lyapunov-like functions and differential inequalities in a fruitful way; (iii) the inclusion of many examples and problems to help the reader develop an expertise in the field. This book is an outgrowth of a seminar course given by the authors. We ix

PREFACE

have assumed the reader is familiar with the fundamental theory of ordinary differential equations, including the theory of differential inequalities, as well as the basic theory of real and functional analysis. It is designed to serve as a textbook for an advanced course and as a research monograph. It is therefore useful to the specialist and the nonspecialist alike. The reader who is familiar with the contents of the book, it is hoped, is fully equipped to contribute to the area.

X

Acknowledgments We wish to express our warmest thanks to Professor Richard Bellman whose interest and enthusiastic support made this work possible. We are immensely pleased that our book appears in his series. m e staff of Academic Press has been most helpfil. We thank our colleagues who participated in the seminar on boundary value problems at the University of Rhode Island in 1971-1972. In particular, we appreciate the comments and criticism of Professors E. Roxin, R. Driver, and M. Berman. Moreover, we gratefilly acknowledge several helpfil suggestions offered by Professor L. Jackson. We are very much indebted to Professors G.S. Lad& and S. Leela for their enthusiastic support in many stages of the development of this monograph and to Mr. T. K. Teng for his careful proofreading. Moreover, we wish to thank Mrs. Rosalind Shumate and Mr. Sreekantham for their excellent typing of the manuscript, and we wish to express our appreciation to Ms.Elaine Barth for her superb vping of the final copy. m e first-mentioned author wouM like to acknowledge some interesting helpfirl discussions on boundary value problems with the differential equation 3 group at the University of Missouri at Columbia. Finally, the final preparation of this book was facilitated by a National Science Foundation Gmnt GP-3 7838.

xi

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Chapter 1 METHODS INVOLVING DIFFERENTIAL INEQUALITIES

1.0

INTRODUCTION A variety of techniques a r e employed i n the theory of

nonlinear boundary value problems.

!Chis chapter i s primarily

concerned with the methods involving d i f f e r e n t i a l inequalities. The basic idea i s t o modify the given boundary value problem suitably, and then t o use the theory of d i f f e r e n t i a l inequalities and the existence theorems i n the small t o establish the desired existence r e s u l t s i n the large. After presenting needed existence theorems i n the small, we f i r s t concentrate on scalar second-order d i f f e r e n t i a l equations and associated boundary value problems.

We then

introduce upper and lower solutions, discuss the modification technique, and u t i l i z e Nagumo's condition t o obtain a p r i o r i bounds on solutions and t h e i r derivatives.

Once we have these

bounds a t our disposal, t o prove existence theorems on f i n i t e o r i n f i n i t e intervals i s r e l a t i v e l y simple and straightforward. Boundary value problems subjected t o nonlinear boundary conditions as well are treated i n t h i s framework.

We then

develop Q,rapunov-like theory f o r boundary value problems employing several w a p o v - l i k e functions and the theory of d i f f e r e n t i a l inequalities i n a f r u i t f u l way.

We also t r e a t

i n d e t a i l Perron's method of proving existence i n the large 1

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

by u t i l i z i n g the properties of sub-

and superf'unctions and

the existence r e s u l t s i n the small.

!Ibis technique works well

f o r scalar equations. We next extend the r e s u l t s considered f o r scalar equations t o 8 f i n i t e system of second-order d i f f e r e n t i a l equations. Here there are two directions t o follow, t h a t is, e i t h e r t r y t o obtain the required bounds componentwise o r i n terms of a convenient norm. We o f f e r r e s u l t s from both points of view indicating t h e i r r e l a t i v e merits and using I&-apunov-like theory, whenever possible, t o derive general r e s u l t s . 1.1 EXISTENCE IN l E 3 SMALL

denote t h e r e a l n-dimensional, Euclidean space

Rn

Let

and f o r x

E

Rn,

16(11

let

denote any convenient norm of

x.

be the i n t e r v a l

[a,b]. We s h a l l mean by C(n)[A,B] the class of n-times continuously differentiable functions Let

J

from a s e t A

into

8.

set

B.

We w i l l be concerned, i n t h i s section, w i t h the existence of solutions of the second-order d i f f e r e n t i a l equations of the form (1.1.1)

x" = P(t,x,x'),

satisfying the boundary conditions (1.1.2) where

n n

f E C[JXRnXR ,R

3.

For the purposes of t h i s chapter,

we also need an existence r e s u l t under more general boundary conditions.

This we do consider f o r the s c a l a r case, l e a v i a

a thorough discussion of the general theory t o a l a t e r chapter. F i r s t of a l l , we observe t h a t the only solution of (1.1.3)

XI1 =

0,

subject t o the boundary conditions 2

1.1. EXISTENCE IN THE SMALL

i s the t r i v i a l solution.

This implies, from the theory of

linear d i f f e r e n t i a l equations, that there e x i s t s a unique solution of

(1.1.5)

h(t),

X" =

satisfying (1.1.4) f o r each h

E

C[J,Rn].

Moreover, since the

problem (1.1.3), (1.1.4) possesses the two l i n e a r l y independent solutions u ( t ) = (t - t l ) J v ( t ) = ( t 2 -t ) , the method of variation o f parameters readily gives the i n t e g r a l equation (1.1.6)

x(t) =

+

-

(t2- t ) (S tl)h(S)

jr2(t -

tl)(t2

- s ) h ( s ) ds

dS

I

for the solution x ( t ) of (1.1.5) subject t o (1.1.4). Relation (1.1.6) can be written i n the familiar form

"1 where

i s usually referred t o as the Green's f'unction f o r the boundary value problem i n question. Hence

%is function G(t,s)

the solution of (1.1.5) verifying conditions (1.1.2) takes the form

3

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(1.1.8)

x(t) =

rt2

G(t,s)h(s) ds

+ w(t),

w(t2) = x where w”(t) = 0 and w(t,) = x 1’ 2‘ follows that if x ( t ) is a solution of (1.1.1), (1-1-9)x ( t )

=lt2

G(t,s)f(S,x(s),

~ ‘ ( 8 ) )ds

It therefore

(1.1.2),

then

+ w(t).

Conversely, i f x ( t ) i s a solution of (1.1.9), we can verify by differentiation of (1.1.6) that x ( t ) s a t i s f i e s ( l . l . l ) , (1.1.2). L e t us next r e c a l l some properties of the function G(t,s) f o r later use. For a fixed t, the maximum of IG(t,s)l i s attained a t s = t and IG(t,t)l has i t s maximum value at t = (t1+t2)/2, that i s ,

(1.1. lo)

Iw,~)I

5 ( t 2 - tl)/J+

Furthermore,

and consequently

Wreover,

-

ds = ( ( t tl)

2

+ (t2 - t ) 2 ) / 2 ( t 2

- tl) ,

the maximum of which is attained a t t = t l and t = t 2 . Hence, we obtain 4

1.1. EXISTENCE IN THE SMALL

We are now ready t o prove an existence and uniqueness r e s u l t by using t h e contraction mapping theorem,

where

K, L

>0

are constants such t h a t

Then the boundary value problem ( l . l . l ) ,

(1.1.2) has a unique

solution. Proof: u E CT[[tl,t2],

Let

B

Rn]

Define the operator Tu(t)

be the Banach space of functions with t h e norm

T: B + B

=Lt2

by

G ( t , s ) f ( s , u ( s ) , u t ( s ) ) ds

1

We then have, by (1.l.l.l)and (1.1.13),

Also, because of (l.l.l2), 5

+ w(t).

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

It then follows t h a t

ThisJ i n view of assumption (1.1.14)' shows t h a t

T

is a

contraction mapping and thus has a unique fixed point which i s t h e solution of t h e problem (l.l.l)J (1.1.2).

m e proof i s

complete.

An i n t e r e s t i n g problem i s t o f i n d the l a r g e s t possible i n t e r v a l i n which t h e preceding theorem i s v a l i d . f E C[JXRXR,R],

when

result.

I n t h e case

one can o f f e r such a best possible

We have i n t e n t i o n a l l y given such a r e s u l t i n t h e

following exercise with generous h i n t s . EXERCISE 1.1.1. Assume t h a t

s a t i s f i e s (1.1.13).

Let

(1.1.15)

U"

which vanishes a t number such t h a t

t

u(t)

and

be any solution of

+Lu' + K u = 0

t

and l e t a(L,K) be t h e f i r s t unique 1 u l ( t ) = O f o r t = t l + a ( L J K ) . Show t h a t t h e =

boundary value problem (1.1.1)' if

f E C[JxRXR,R]

(1.1.2) has a unique solution

t 2 - t l < 2a(LJK) and t h a t t h i s r e s u l t i s best possible. Hints:

Step 1. F i r s t show t h a t t h e r e i s a unique solution

t o t h e boundary value problem x 1 ( t 3 )=

3

if

XI'

-

(t3 tl) < a(L,K).

= f(t,x,x'), x(t ) = x 1 1'

This can be shown by apply-

ing the contraction mapping theorem r e l a t i v e t o the M a c h space

E = C(1)[[tl,t3],R]

with t h e norm 6

1 . 1 . EXISTENCE IN THE SMALL I-

where u o ( t )

>0

i s a solution of

for

Lu' +Ku =

+

UI'

cy

s u f f i c i e n t l y close t o 1.

cy

Step 2.

Show t h a t the existence of unique solutions of

(l.l,l),(1.1.2) and of (1.1.1)with e i t h e r

x ( t l ) = x 1' ) = x 3 ' x ( t 2 ) = x 2 on any i n t e r v a l of

x'(t ) = x or x ( 3 3 ' length l e s s than

implies the existence of a unique solution

d

of (1.1.1)' (1.1.2) on any i n t e r v a l of length l e s s than Step 3. show t h a t ing

2d.

To prove t h a t the r e s u l t i s the best possible

u" + Llu'

I + Ku = 0

u(t,) = u ( t 2 )= 0,

where

has a n o n t r i v i a l solution verif'y-

t2- t l =2CY(L,K).

Since u ( t ) = O

also s a t i s f i e s the problem, argue t h a t the r e s u l t i s best possible.

Observe t h a t

EXERCISE 1.1.2.

i n Theorem 1.1.1f o r where N

m

a(L,K)

can be e x p l i c i t l y computed.

Show t h a t it i s s u f f i c i e n t t o define

t

E

/(XI( -< N,

[tl,t21y

IIx' 11 < 4N/(t2

f

- tl),

satisfies either

2 ( t 2 tl>

-a

-< N [ l - ( K

Hint: Apply hll05 N where

2 ( t 2 tl>

-a

t h e contraction mapping theorem on the b a l l

7

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

To obtain merely an existence result we employ Schauder's

fixed point theorem as is usual.

THEOREM 1.1.2. l e t , f o r t E J, IlxII where

f E

I& M > 0, N

ItYII ,< PJ,

5 m,

C[JFRnXRn ,Rn 1.

>0

be given numbers and

Ilf(t,x,Y)II

5

q,

Suppose that

6 = 1nin[(8M/q)~, 2N/ql. Then any problem (1.1,1), (1.1.2) such that [t1,t21 J, t2 tl I 6, Ilx1II 5 M, I ~ x, I 5 M, and Ilx, x211/(t2 t l ) I N, has a solution. Furthermore, given any E > 0, there i s a solution x ( t ) such that

-

-

-

llx(t) -W(t)ll < E, Ilx*(t) -wl(t)ll t2 tl i s sufficiently small.

-

is a closed, convex subset o f

bY

ICx(t)

=

< E on [tl,t21, provided

B. Define the mapping T: B+B

1"'

G(t,s) f (s,x(s),xl (s)) ds + w(t).

Using now estimates (l.l.ll), (l.l.E!),

8

we obtain

1.l.EXISTENCE IN

THE SMALL

and

t2- t l 5 6, T maps

Hence, f o r

II(W"(t)Il

5

into i t s e l f .

Bo

Ilf(t,x(t), x ' ( t ) ) l l

by Ascoli's theorem it follows t h a t

T

5

Also, since

q,

i s completely continuous.

Schauder's fixed point theorem therefore assures t h a t fixed point i n If

x(t)

which i s a solution of ( l . l . l ) ,

Bo,

x

IIx(t) - w ( t )

t

E

(1.1.2).

i s a solution of t h e boundary value problem

(l.l.l), (1.1.2), with

for

T has a

[tl,t2].

E

Bo,

then we have

-

II 5

( ( t 2 tl12/8) 9,

Consequently t h e last assertion follows and

the proof of Theorem 1.1.2 i s complete. COROLLARY 1.1.1. Assume t h a t

and i s bounded on

[tl,t2] XRn xRn.

f E C[[t,,t2]

x R n xRn,Rn]

Then every boundary value

problem (l.l.l), (1.1.2) has a solution. Proof:

where

q

Pick M

>0

s u f f i c i e n t l y large so t h a t

i s an upper bound of

f.

The conclusion then follows

from Theorem 1.1.2.

As was indicated e a r l i e r , we s h a l l next discuss the 9

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

s i t u a t i o n where f E C [J XR x R , R ] . Then a natural question is under what general boundary conditions can one construct a Green's m c t i o n f o r the scalar equation (1.1.3). L e t us look a t t h e boundary conditions

- a2x1 (t,)

(1.1.16)

alx(tl)

(1.1.17)

b x ( t 2 ) +b2X1( t 2 ) = 0 .

= 0,

1

A s long as there e x i s t two l i n e a r l y independent solutions of

(1.1.3) satisfying (1.1.16), ( l . l . l 7 ) , it is possible t o cons t r u c t the desired Green's f'unction. One sufficient condition i s t o f i n d two nonparallel l i n e s satisfying (1.1.16) and (1.1.17), t h a t i s , demanding t h a t alb2 + a2bl

(1.1.18)

#

0.

Another s u f f i c i e n t condition, for example, i s t o require t h a t a2b2 = 0 . I n view of these remarks, it is s u f f i c i e n t t o assume that a1+ b1 > 0,

2 0,

(1.1.19) al,a2,bl,b2

and

a2+b2

> 0.

L e t us now consider the nonhomogeneous boundary conditions (1.1.20)

alx(tl)

-

(1.1.21)

blx(t2)

+ bF'(t2)

a2x1(tl) = A, = B.

It i s not difficult; t o conclude the existence of a solution cp(t) satisfying (1.1.3) and (1.1.20), (1.1.21). Thus any

solution x ( t ) of (1.1.1)obeying the boundary conditions (1.1.20) and (1.1.21), takes the form (1.1.22)

x(t)

Go(t,s)f(s,x(s>,x(s>)

= Jt2

1 where 10

+ T'(t),

1.1. EXISTENCE IN THE SMALL

and

- v(t)u'(t).

c = u(t)v'(t)

Here we assume u ( t ) , v ( t )

are two l i n e a r l y independent solutions o f (1.1.3)which s a t i s f y

(1.1.16), (1.1.17). This discussion leads t o t h e following result. TEEOREM

1.1.3.

[tl,t2] X R X R . (1.1.20),

f E C [[t,,t2]

Let

X R xR,R],

bounded on

Then t h e boundary value problem (l.l.l),

(1.1.21) has a solution, whenever (1.1.18) o r (1.1.19)

holds. Proof:

Let

Define a mapping

M be the bound of T: E + E

where the Banach space

f

on

[tl,t2] x R X R .

by

E = C ( l ) [[tl,t2],R]

Letting

it follows t h a t 11

with t h e norm

1. METHODS INVOLVING DIFFERENTIAL INEQUALITiEs

Hence,

T maps the closed, bounded, and convex s e t

Bo = [X E E: ( x ( t )I 5 NM+L,

Ix'

(t) I

5 NIM+L1]

into i t s e l f . mreover, since I ( ~ ) " l < M, T i s completely continuous by Ascoli's theorem. The Schauder's fixed point theorem then yields the fixed point of T w h i c h i s a solution of (l.l.l), (1.1.20), (1.1.21), thus completing the proof of' the theorem. EXEXCISE 1.1.3.

-

Find the solution of the boundary value

problem x" = t, x(0) x' (0) = 1 and x(1) = 0. 1.2 UPPER AND lxlwER SOLUTIONS Let us consider the second-order equation

(1.2.1)

x" = f(t,X,X'),

where f E C[JxRxR,R], J being the i n t e r v a l [a,b] as before. The i n t e r i o r of J w i l l be denoted by J 0 L e t us define certain types of solutions of d i f f e r e n t i a l inequalities t h a t w i l l play a prominent part i n the subsequent work.

.

DEFINIITION 1.2.1. i s said t o be

A f'unction

cy E

C[J,Rl

(i) a lower solution of (1.2.1) on

-

D "'(t) =lim inf

h+ 0

(ii)

TI

a' ( t + h ) -a'( t - h ) 2h

J

h-t 0

-

C(l)[Jo,R]

if

,> f ( t , a ( t ) , a ' ( t ) ) ,

upper solution of (1.2.1) on J

D-a'(t) slim sup

n

if

a t ( t + h ) - a t ( t h) < f(tp(t),a'(t)), 2h

-

A f'undamental r e s u l t concerning the upper and lower 12

t cJo;

0

t EJ

1.2. UPPER AND LOWER SOLUTIONS

solutions i s t h e following.

THEOmM 1.2.1. (i) f

in x

C[JxRxR,R]

E

f o r each (ii)

Assume t h a t and

(t,y) E J x R ;

i s a lower solution and

cy

i s nondecreasing

f(t,x,y)

i s an upper solution

@

of (1.2.1) on J ; (iii)

@ ( a )5 @ ( a ) and cy(b)

5 B(b).

If one of t h e d i f f e r e n t i a l i n e q u a l i t i e s involved i s s t r i c t ,

then we have Proof: some t E J that

m(t)

sequently,

a(t)

0

< B(t) on J 0

.

Suppose, on t h e contrary, t h a t

.

@ ( t )2 B(t)

Then, s e t t i n g m(t) = a ( * ) - B ( t ) ,

m(to)

> 0,

m ' ( t o ) = 0,

we notice

to E J

has a nonnegative maximum q t some

0

D-mt(tO) 5 0.

and

for

.

ConHence

Thus

i n view of t h e monotonic character of

f.

!this i s a contra-

diction since one o f t h e d i f f e r e n t i a l i n e q u a l i t i e s i s assumed t o be s t r i c t .

The proof i s therefore complete.

We observe t h a t the proof of Theorem 1 . 2 . 1 breaks down i f one of t h e d i f f e r e n t i a l i n e q u a l i t i e s i s not assumed s t r i c t . Nonetheless, t h e following assertion i s t r u e , DIEOREM 1.2.2. 1 . 2 . 1 hold.

Assume -theses

Suppose f'urther t h a t 13

( i ) - ( i i i ) o f Theorem f(t,x,y)

obeys a one-sided

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Lipschitz condition i n y

on each compact subset of

J X R xR,

that i s ,

Then the inequality a ( t ) _ p ( t )

on J .

U(d)

- B(d)=

f o r some t

[c,d]

Let

a(t)

E,

i s true.

C

E

J

0

.

Let

be such t h a t

J

- p(t) 2

t

E,

E

[C,dI

-

to E (c,d) such t h a t a ( t o ) p ( t o ) = 2 E . Consider the compact subset given by

and there e x i s t s a

s

=

[(t,x,y): t

For any

6

E

[c,dl,

IPW-xl

5

1, I B ' W - Y I

_< 11.

> 0 s u f f i c i e n t l y small, l e t p ( t ) s a t i s f y the

conditions p" = ( L + l ) p ' ,

-min(6,E)

 P ( t ) ,

t

E

(c,d).

m(t)

Moreover, f o r

using successively the assumptions on If we now l e t

t

v ( t ) = m ( t ) + E,

t

E

[c,d],

and

(c,d),

f.

we see, because of the

f,

D'v'(t) =D'm'(t) < f ( t , m ( t ) , m ' ( t ) ) _ < f ( t , v ( t ) , v ' ( t ) ) , Also, 14

t

E

(c,d).

1.2. UPPER AND LOWER SOLUTIONS

+

E

-< m(c) +

E =

v(c),

a ( d ) = @ ( d )+

E

< m(d) +

& =

v(d).

a(.)

= B(c)

An a p p l i c a t i o n of Theorem 1 . 2 . 1 y i e l d s t h a t a(t)

< v(t),

t

E

(c,d).

we o b t a i n

From t h e d e f i n i t i o n of m(t),

B(t) _ < m ( t ) and, as a r e s u l t , we o b t a i n a t

B(t) +

E

t=t

0'

This contradiction proves t h e theorem.

We can deduce a uniqueness r e s u l t f o r a c e r t a i n boundary value problem from Theorem 1.2.2. COROLLARY 1.2.1.

L e t t h e assumptions of Theorem 1.2.2

hold except ( i i ) and ( i i i ) . are s o l u t i o n s on x(t2) = y(t,).

[tl,t2]

with respect t o

y

i s no longer v a l i d .

x"=($)

C

Then x ( t )

REMARK 1.2.1.

This we s t a t e as a c o r o l l a r y .

Suppose t h a t

x,y

such t h a t

x(t,)

J

E

y(t)

on

E

C(2)[[t,,t2],R]

= y(tl),

[tl,t21.

If t h e Lipschitz condition on f ( t , x , y ) on compact sets i s omitted, Corollary 1 . 2 . 1 For example, t h e boundary value problem

(: 5

has s o l u t i o n s x ( t )

1 and

x ( t ) = lt15/2 on

[-1,11.

Instead o f the one-sided Lipschitz condition on f, one could assume t h a t s o l u t i o n s of i n i t i a l value problems XI'

= f(t,x,x')

Theorem 1.2.2.

a r e unique, t o conclude t h e v a l i d i t y of This i s p r e c i s e l y what t h e next theorem does.

THEOREM 1.2.3.

Suppose t h a t 15

I . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(i) in

x

f

C[JXRXR,R]

E

f o r each (ii)

(tJy)

E

and

f(tJx,y)

i s a lower solution and

CY

i s nondecreasing

JxR; i s an upper solution

@

of (1.2.1) on (iii) are unique;

[c,d] C J; solutions of i n i t i a l value problems f o r (1.2.1)

(iv) a ( c >< B(c)

a(t)

Then we have

and CY(d-1 5 @ ( a ) .

5 p(t)

on

[CJd].

The proof of t h i s theorem r e s t s on a r e s u l t t h a t we are going t o consider i n t h e following section and therefore w i l l be given t h e r e . COROLLARY 1.2.2.

If

f(t,x,x')

s a t i s f i e s hypotheses ( i )

and ( i i i ) of Theorem 1.2.3, and i f solutions on

[tl,t2]

x ( t 2 ) = y ( t 2 ) J then

C

J

x,y

such t h a t

E

C(2)[[tl,t2]JR]

are

x(tl)=y(tl),

x ( t ) = y ( t ) on

[tl,t21.

Another i n t e r e s t i n g r e s u l t concerning lower and upper solutions which would y i e l d uniqueness of solutions with general l i n e a r boundary conditions i s t h e following. THEOREM 1.2.4.

(i)

f E C[JxRxR,R]

creasing i n (ii)

Assume t h a t

x f o r each CY

and

(t,y)

E

i s s t r i c t l y in-

f(t,x,y) JxR;

i s a lower solution and

@

i s an upper solution

of (1.2.1) on J; (iii)

f o r each

(t,x,yl),

If(tJx,yl)-f(tJxJy2)I (iv)

a,a(a>

(t,x,y2) E J X R X R j

5

LIY1-Y21J

- a2a' ( a ) 5 al@(a) - a2@'(81,

bl,(b)+b2ct'(b) 5 bl@(b)+b2@'(b), where b +b2 1

> 0,

>O;

a1,a2,blJb2

20,

and

16

a +bl 1

a 1+a2 > 0 ,

>o.

1.2. UPPER AND LOWER SOLUTIONS

Then a ( t ) Proof: -

5

p(t)

on J .

We s h a l l f i r s t show t h a t at the end points of

J

the desired inequality holds. and hence

that

> B(a) which implies from ( i v ) t h a t a2 # 0 a' (a) > p ' ( a ) . Then t h e r e e x i s t s a 6 > 0 such > p ( t ) on [a,6). Consequently, using t h e s t r i c t

.(a)

Let

a(t)

monotony of

and condition ( i i i ) , we obtain

f

D

-

0'

- D-p'

(t)

(t)

-

_> L [ a ' ( t ) B ' ( t ) ] ,

which, by t h e theory of d i f f e r e n t i a l inequalities, y i e l d s a' ( t )

- B ' ( t ) 2 [a'( a ) - B ' ( a ) le L(t-a)

This, together with a ' ( t ) U'(t)

p'(t)

on

[a,6].

- p ' ( t ) -< 0,

assures us t h a t

I n view of (1.2.2), we then a r r i v e

a t the contradiction

-

D-a' ( t ) D-B' ( t ) > 0 which establishes t h e claim

to E [a,s).

(to)

-

- Bf(t) > 0

argument may be extended t o show t h a t a(b)

>0

f o r some

Again, we appeal t o t h e d i f f e r e n t i a l inequality

(1.2.2) t o conclude a ' ( t ) that

(to)

> p(b).

on 6= b

[t0,6].

This

which implies

The l a s t conclusion, i n i t s turn, leads

t o the contradiction a l ( b )

-

p(a)

i s impossible.

a ( t ) 5 B(t)

a ( & ) > p(a)

Thus

a(b)

If we assume t h a t

>

i s impossible.

p(b), we a r e lead t o t h e re-

arguing as before, which we have shown

It therefore remains t o be shown t h a t

on

which i s exactly s i m i l a r t o t h e proof

(a,b),

of Theorem 1.2.1.

The proof of t h e theorem i s therefore

complete.

1.3 THE MODIFIED FUNCTION Employing the notion of upper and lower solutions and t h e existence theorem i n t h e small, it i s possible t o e s t a b l i s h t h e existence of solutions i n t h e large of some boundary value problems f o r a modified form of the d i f f e r e n t i a l equation XI' = f(t,X,X'),

(1.3.1) where

f E C[J xRxR,R].

Let us f i r s t define the modified

function. DEFINITION 1.3.1. on

J

and l e t

c

J.

Then define

>0

Let

a,p

C(l)[JJR]

E

be such t h a t

with

Icy'(t) I, I p l ( t )

I

a(t) < p(t)

< c on

2 c,

f(t,x,c)

for

XI

f(t,X,X')

for

IX'I

f(tJx,-c)

for

x'

5 CJ 5 -c,

and

F(t,x,x') =

F{ The function

F*( t ,B (t),XI) + [x-B(t) ] / 1 + x2

for

F*(tJx,x')

for a ( t ) < x < B ( t ) , for x < a ( t ) .

* (t , a ( t ) , x ' ) + [ x - a ( t ) ] / 1 + x 2

F(t,x,x')

J

w i l l be called the modification of

f ( t J x J x ' ) associated with t h e t r i p l e follows from the d e f i n i t i o n t h a t on J x R x R

x >p(t)

and t h a t 18

a ( t ) , p(t), c.

F(t,x,x')

It

i s continuous

1.3. THE MODIFIED FUNCTION

Let us consider now the modified boundary value problem

(1.3.2)

X"

= F(t,X,X'),

.(a)

=

y, x(b)

= 6,

r e l a t i v e t o which we have t h e following.

THEOREM 1.3.1.

a,f3 E C(l)[J,R]

Let

be, respectively,

lower and upper solutions of (1.3.1) on J a(t)

5

B(t)

on J .

such t h a t

Then t h e boundary value problem (1.3.2),

i s the modification of f associated with t h e t r i p l e cr(t)J B(t)J c, has a solution x E C(2)[J,R] s a t i s f y i n g where

F

a ( t >5 x ( t )

(1.3.3) provided t h a t

Proof:

5

.(a)

y

B(t)

on J,

-

p(a), a ( b ) < 6

5

B(b).

By Corollary 1.1.1, t h e boundary value problem

(1.3.2) has a solution show (1.3.3).

5

5

x c c ( 2 ) [ ~ , ~ ~~ h. u swe only need t o

We s h a l l only prove t h a t

x(t)

5

@ ( t ) on J .

The arguments are e s s e n t i a l l y t h e same f o r t h e case a(t)

x(t).

Assume, i f possible, t h a t

-

x(t)

> B(t)

f o r some

t E J . Then x ( t ) p ( t ) has a positive maximum a t a point 0 to E J Hence it follows t h a t x ' ( t , ) = @ ' ( t o ) , I x ' ( t O ) ( c c

.

and

x"(to) = F ( t O , x ( t O ) , x '( t o ) )

19

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and therefore, we arrive at

-

which i s impossible a t a maximum of x ( t ) B(t). We conclude that x ( t ) 5 B(t) on J . The proof i s complete. Let us next consider the general linear boundary conditions

(1.3.4)

a,x(a)

-

(a) = A,

blx(b)

+ b2x1(b)

= B,

where a1 + a2 > 0, b 1 + b2 > O j a1ja2jbljb2 2 0, and a + b l > 0 . We w i s h t o prove a result analogous t o Theorem 1 1.3.1 w i t h respect t o conditions (1.3.4). F i r s t of a l l , we need the following lemma.

LFMMA 1 . 3 . 1 . Assume that ( i ) a,@ E C(l)[J,R] are, respectively, lower and upper solutions of (1.3.1) on J such that a ( t ) 5 B(t) on J; ( i i ) f E C[J XRXR,R] and for t E Jj a ( t ) 5 x 5 B ( t ) , f ( t , x , x ' ) s a t i s f i e s a Lipschitz condition i n X I f o r a constant L > 0. Then, there exists a f'unction f* E C [ J xRxR,R] which is bounded on J X R X R and i s Lipschitzian for the same constant L > 0 whenever t E J and a ( t ) 5 x ,< B(t).

Proof:

A s before, l e t

I a ' ( t ) l , lB'(t)l t E J and a ( t )

c > 0 be such that

< c on J. Then define f*(t,xJx') f o r 5 x 5 p ( t ) by setting

20

1.3. THE MODIFIED FUNCTION

f*(t,x,x*) =

{

x ' 2 c, ( x ' ) 5 c,

f(t,x,c) f(t,x,xl)

for for

f(t,x,-c)

for

We then extend the domain of definition of J X R X R by l e t t i n g f*(t,x,x')

XI

5 -c.

f*(t,x,x')

f*(t,B(t),x')

for x > B(t),

f*(t,a(t),x')

for x < a ( t ) .

to

=

It i s easy t o see that the function f* so defined possesses a l l the stated properties. Hence the proof i s complete. E-EOREM 1.3.2.

Let hypotheses ( i ) and ( i i ) of Lemma 1 . 3 . 1

Then there exists a solution x

hold.

(1.3.5)

E

C(2)[J,R]

of

XI' = f*(t,X,X')

which s a t i s f i e s the boundary conditions (1.3,4), provided that

(1.3.6)

.,.(a)

Proof: F(t,x,x') =

- a2a' (a) 5 A 5 alp(") - a 2 P (a),

Define the ftmction F(t,x,xl)

{

on J x R x R as

f*C(t,B(t),xl) + ~ ( [ x - ~ ( t ) ~ / l +for x ~ )x > ~(t), f*(t,x,x') for 4 t ) _ < X S B ( t ) , f*(t,a(t),x') + ( [ x - a ( t ) ] / 1 + x 2 ) f o r x < a ( t > ,

where f*(t,x,x') i s the f'unction obtained i n &ma 1.3.1. Since f* is bounded, F is also bounded. Hence, by Corollary 1.1.1, there exists a solution x E C ( 2 ) [ J , R ] of (1.3.8)

X"

= F(t,x,x')

satisfying the boundary conditions (1.3.4).

21

We now show that

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

t

for

E

J, a ( t )

-< x ( t ) -< @ ( t ) ,

which implies t h a t

a solution of the problem (1.3.4), (1.3.5). Assume t h a t x(a) > @(a) i n which case zero.

Then there exists a

[a,6).

By definition of

(1.3.9)

x"(t) - D - B ' ( t )

f*,

+

f*(t,B(t),x'(t))

-

cannot be

h(t) - B(t)l 2 l + x (t)

f*(tJ@(t)?B'( t ) )

-> -Llx' ( t ) - B ' ( t ) I since

Y

i s a l s o an upper solution of (1.3.8).

p(t)

is

> a such t h a t x ( t ) > p ( t ) on F, we have f o r a
6

2

a2

x(t)

Also,

r e l a t i o n (1.3.6) implies, because of the assumption x ( a ) _> p(a), t h a t x ' ( a ) 2 @ ' ( a ) . We claim that there i s a to E [a,6) such that

x'(t,)

on a

5

t

5

- @ ' ( t o )> 0 .

6.

If not, suppose t h a t

Then, by (1.3.9), x"(t)

- D-B'

x'(t)

- p'(t)IO

we would obtain

( t ) _> L[X' ( t )

- B' (t)1

which, by the theory of d i f f e r e n t i a l inequalities, yields x'(t)

- B'(t) 2

This, together with on

a
x'(t)

[x'(a)

- B'(a)le L(t-a)

- B ' ( t ) -4,

implies t h a t

x ' ( t ) Epl(t)

On the basis of (l.3.9), we then arrive a t the

contradict ion

-

x ' ( t o ) @ ' ( t o )> 0 f o r some to E [a,&). Again appealing t o the d i f f e r e n t i a l inequality (1.3.9), we conclude t h a t x ' ( t ) - B ' ( t ) > 0 on [t0,8]. I n

which establishes the claim,

fact, t h i s argument leads t o follows t h a t tradiction

x(b) > p(b).

6 = b and, consequently, it This, i n turn, leads t o the con-

< @'(b), i n view of the relations (1.3.4) x'(b) -

and (1.3.7). 22

1.3. THE MODIFIED FUNCTION

I f , on the other hand, we assume t h a t are lead t o the r e l a t i o n

x(b) > B(b), we arguing as before, which

x(a) > B(a)

has j u s t been shown t o be impossible. show that

x(t)

5

t

@(t) for

Assume then there e x i s t

E J

t

x(t)

- p(t),

It therefore remains t o

.

f o r which

E Jo

tl,t2

"his implies the existence of

The difference

0

E

J

> B(t).

x(t)

such t h a t

therefore, assmes a positive

maximum a t some to E (tl,t2). Clearly, x ' ( t o ) = @ ' ( t o ) and x"(to) D-B' ( t o ) _< 0 . A computation, however, leads t o the contradiction

-

noting ( x t ( t o l = l p T ( t o ) l< c and the definition of f". We < B(t) on J . One can similarly have thus established x ( t ) verify that

a(t) < x(t)

on J

completing the proof.

Let us now proceed t o supply the proof of Theorm 1.2.3.

For t h i s purpose we need the following r e s u l t . THEOREM

1.3.3. Assume t h a t

f E C[JxRxR,R]

and t h a t

,

solutions of i n i t i a l value problems f o r (1.3 .l)are unique. Let a,@ E C(l)[J,R] be lower and upper solutions o f (1.3.1) on J, respectively, such t h a t a ( t ) < p ( t ) on J . Then, i f a ( t o ) = @ ( t o ) and CY ( ) = p ( ) f o r some to E J, it follows ' ' that a ( t ) p ( t ) on J .

Proof:

Suppose t h a t

a ( t ) f p(t)

the case where there i s an i n t e r v a l

on J .

(to,tl]

C

We w i l l consider J

such t h a t

a ( t o ) = @ ( t o ) ,Q ' ( t o ) = B ' ( t o ) , and a ( t ) < B(t) on 23

[to,tlI-

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Fl(t,x,x')

Let

be the modification jknction of

according t o Definition 1.3.1 f o r the i n t e r v a l the t r i p l e a,p,cl.

f(t,x,x') [to,t,]

and

a ( t l ) < S1 < @(tl), we have by

Then, i f

Theorem 1.3.1 t h a t the boundary value problem (BVP) =

X"

Fl(t,x,x'),

has a solution x1 a(t)

5 x,(t) _<

E

B(t)

x(to) = a ( t o ) C(2)[[to,t,],R] on

[tO,tll.

x ' ( t o ) = a l ( t o ) and since of

such that Therefore,

I x i ( t ) I < c1

to, it follows by the definition of

interval

[t0,t2]

(1.3.1).

If

C

t2 = t

[tO,tl]

x ( t 1) = 61

xl(tO)

=

u(to),

on some neighborhood

F1, there i s a maximal

on which x ( t ) 1

i s a solution of

1'

a ( t 2 ) = a@,) < S1 = Xl(t2) = x 1(t1) < @ ( t l ) .

to < t 2 < t

it i s s t i l l t r u e t h a t m(t2) < xl(t2) < p(t2) 1' for, if e i t h e r inequality were an equality, we would have

If

Ixi(t)l

< c1 and the i n t e r v a l [t0 , t 2 3 would not be maximal.

We can therefore construct another modification f'unction

[t ,t ] r e l a t i v e t o the t r i p l e 0 2 a , p, c2. A n application of meorem 1.3.1 again w i t h a ( t 2 ) < 62 < p ( t 2 ) , shows t h a t there i s a solution x2 E C(2) [[t0,t2],R] of the BVP F2(t,x,x')

of

f(t,x,x')

satisfying

a ( t ) < x2(t)

-

on

5 p(t)

on

[t

0'

t 2 1.

it follows t h a t there i s a maximal i n t e r v a l on which x 2 ( t )

A s before,

[to,t3] i s a solution of (1.3.1) and t h a t

C

[t0,t21

a ( t 3 ) < x2(t3) < p(t3). lnnis contradicts the assumption that solutions of i n i t i a l value problems f o r (1.3.1) are unique, since xl(tO) = x 2 ( t 0 ) and x l ( t ) = x;(to). conclude t h a t

a(t)

p(t)

1

on

24

0 J.

We therefore

1.4. NAGUMO'S CONDITION

We a r e now ready t o give t h e proof o f Theorem 1.2.3. Proof of meorem 1.2.3: some points o f

[c,d]

Assume t h a t

and l e t

Clearly t h e r e e x i s t s a

a(t)

-

> p ( t ) at

N = m a x [ a ( t ) @ ( t ) l on

-

[c~d].

to E [ c , d ] where a ( t O ) @ ( t o )= N

and a ' ( t o ) = B ' ( t o ) . The functions (Y and B , = p + N s a t i s f y t h e conditions o f Theorem 1.3.3 i n view of t h e nondecreasing nature f(t,x,xl)

of

@ ( t )+ N

a(t)

in

Consequently it follows t h a t

x.

on

[c,d],

a c o n t r a d i c t i o n which proves t h e

theorem.

1.4 NAGW'S CONDITION A s we have seen, t h e proofs f o r t h e existence theorems

for solutions o f boundary value problems depend on f i n d i n g a Here we

p r i o r i bounds f o r t h e s o l u t i o n and i t s d e r i v a t i v e .

present some s u f f i c i e n t conditions f o r obtaining such bounds. DEFINITION 1.4.1. C[J,R]

with

Let

E

t

J, w ( t ) _< x _< @ ( t ) and

(1.4.1) where h

If(tJXYX') E

C[JXRxR;R]

E

a(t) < @ ( t ) on

a,p E

f

x'

E

J.

and

Suppose t h a t f o r

R,

I 5 h ( Ix' 1 )

C [R+j ( o J m ) ] . If

THEOREM 1.4.1. condition on J

Assume t h a t

f

satisfies Nagumo's

with respect t o t h e p a i r 25

a , @ . m e n f o r any

1.

METHODS INVOLVING DIFFERENTIAL INEQUALITIES

solution x E C(')[J,R] of (1.3.1) with a ( t ) _< x ( t ) ,< @ ( t ) on J, there e x i s t s an N > 0 depending only on a,@,h such that Ix'(t) I

(1.4.4)

5N

on J .

Because o f (1.4.2), we can choose an N > h such

Proof: that

(b - a ) x l ( t o ) = x(b)

to E Jo is such t h a t

If

- x(a),

then by

I x l ( t O )I 5 h. Assume t h a t (1.4.4) i s not Then there e x i s t s an i n t e r v a l [tl,t21 C J such t h a t

(1.4.3), we have true.

the following cases hold: = N,

and

A < x'(t)

=

N, x l ( t , ) = A,

and

h

x ' ( t 1)

=

-A,

x'(t,)

= -N,

(i) x l ( t ) = A, x ' ( t , ) 1

< N,

t c (tl,t2),

( i i ) x'(t,)

t

E

(tl,t2), (iii)

t

E

(tl,t2), (iv)

t

E

< x ' ( t ) < N,

x'(t2)

= -N,

and

-N < x ' ( t )

< -A,

x

=

-h,

and

-N < x l ( t )

<

)

-A,

(tl,t2).

L e t us consider case ( i ) .

On

[tl,t21,

and as a r e s u l t

26

we obtain, by (l,b.l),

1.4. NAGUMO'S CONDITION

This leads t o t h e c o n t r a d i c t i o n

We can d e a l with t h e remaining p o s s i b i l i t i e s i n a s i m i l a r way

and therefore we conclude t h a t (1.4.4) i s v a l i d .

Then, f o r any s o l u t i o n exists an N lx'(t)l

5

>0 on

N

x

E

C(2)[J,R1 with

depending only on J.

Also,

N -10

M,h, (b

as

( x ( t ) ( 5 M,

- a)

there

such t h a t

M -0.

The conclusion of Corollary 1 . 4 . 1 i s n n C[JxRnxR , R ] and absolute values a r e re-

REMARK 1.4.1.

false, i f

f

E

placed by norms, as t h e following example shows. (cos n t , s i n n t ) llX"

( t ) 11 = n2 =

hold with

(Ix'

so t h a t ( t ) I/*.

Let

x(t)

=

Ilx(t)(I = 1, Ilxl(t)ll = n, Thus t h e assumptions of Corollary 1 . 4 . 1

M = 1, h ( s ) = s2

+

1. However, t h e r e does not e x i s t

an N > 0 such t h a t / / x t ( t ) )_<) N f o r a l l choices of n. Thus it is c l e a r t h a t Nagumo's condition i s not s u f f i c i e n t t o obtain the desired conclusion i n t h e case of systems. We s h a l l discuss t h i s f u r t h e r when we t r e a t systems of equations l a t e r i n t h i s chapter. 21

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

EXERCISE 1.4.1.

on

Suppose t h a t

J.

a(t)

Let

5

5

where

L

>

f E C[JxRxR,R]

p(t), Y 1 J Y 2

I 0

(

J xJ

C(l)[JJR] with

E

Y,)

and f o r

a(t)

t

E

-< p ( t )

JJ

RJ

-

(t J xJY2)

i s a constant.

Show

condition. THEOREM 1.4.2.

Assume t h a t such t h a t

( i ) a,p E C(l)[J,R] a(b) = p(b)J W

and a ( t )

5

p(t)

On

.(a)

= p(a),

J;

( i i ) (pJJI E C(')[uJR] with q ( t J x ) 5 Jl(t,x) on = [(t,X): a ( t ) <X < (u(t)J t E J ] and O n W,

(1.4.5)

q t ( t J x ) + qx(tJx)q(tJx)

< f(tJxJq(tJx)),

(1.4.6)

$ t ( t J x ) + qx(tJx)q(tJx)

> f(tJXJq(tJX)),

Proof:

a(t)

Let

x(t)

(u,

where

be any solution of (1.3.1) such t h a t

_< x ( t ) 5 p ( t ) on J . Then, by

(i)and ( i i i ) it follows

that ' ~ ( a , x ( a )5 )

X I(a)

Assume now t h a t there exists a

5

Jr(a,x(a)).

to E (a,b)

such t h a t

< ' p ( t o J x ( t o ) ) . Then there must e x i s t an i n t e r v a l [tl,t2]C J with tl < to < t2 such t h a t x ' ( t i ) = ' P ( t i > x ( t i ) ) Ji = 1,2 and x ' ( t ) < ' P ( t , x ( t ) ) , t i < t < t 2 J

x'(to)

28

1.4. NACUMO'S CONDITION

and f u r t h e r t h a t

This i s a c o n t r a d i c t i o n because of assumption ( i i ) .

f o r e conclude t h a t merit shows t h a t

x'(t)

x'(t)

-> ' p ( t , x ( t ) )

-< J r ( t , x ( t ) )

on

on J

We there-

A s i m i l a r argu-

J.

and t h e proof i s

complete. We remark t h a t t h e i n e q u a l i t y s i g n i n (1.4.5) could be reversed i n which case t h e c o n t r a d i c t i o n i s a r r i v e d a t because of

(1.4.6).

A s i m i l a r coment holds r e l a t i v e t o i n e q u a l i t y

Of

course, i n t h i s case, we have t o replace ( i i i ) by V(b,a@))

5 a'@>,

_< Jr(b,B(b))*

COROLLARY 1.4.2. Assume ( i ) of Theorem 1.4.2.

e x i s t constants B'(a),B'(b)

N

5N2,

N such t h a t 1 ' 2 and f(t,x,Ni)

a"(a), a' (b)

#

0, i = 1,2.

clusion of Theorem 1.4.2 i s t r u e w i t h $ { t A=

9.

EXERCISE 1.4.2. Assume t h a t

Ix'I

--fw

f

E

Let there

2 N1, Then t h e con-

cp(t,x) = N1

and

L e t hypothesis ( i ) of Theorem 1.4.2 hold.

C[JXRXR,R]

and

uniformly on compact

lf(t,x,x')I

(t,x)

sets.

-tm

as

Show t h a t t h e

conclusion of Corollary 1.4.2 i s t r u e . EXERCISE 1.4.3. 1 . 4 . 1 h o l d and flmctions

h(s)

Show t h a t i f t h e assumptions o f Corollary i s nondecreasing, t h e r e always e x i s t

'p(t,x), @ ( t , x ) (independent of

t ) satisfying the

hypotheses of Theorem 1.4.2. Consider t h e ' following example on J = [-1,11,' 29

1 , METHODS

= e-2(x+1)

XI'

where

n

INVOLVING DIFFERENTIAL INEQUALITIES

- (x')2nJ

x(-1) = 0 = x ( l ) ,

i s a p o s i t i v e integer.

B ( t ) = 0.

Choose N1 = - 2

Take

and

a ( t ) = t 2 - 1 and

N2 = 2.

Then we see a l l t h e

hypotheses of Corollary 1.4.2 a r e s a t i s f i e d . t h a t any solution x ( t ) Ixr(t)(5 2

on J.

such t h a t

Hence we conclude

t 2 -15 x ( t ) _< 0

However, we notice t h a t f o r n

satisfies

>

* J

Corollary 1 . 4 . 1 is not applicable. A variant of Theorem 1.4.2 which i s more useful i s t h e following r e s u l t .

Proof:

Let

f (t>XJX') such t h a t

o(t)

_< x ( t ) 5 B(t), t

to E (a,bl

be any solution o f

x E C(2)[J,R1

such t h a t

cp(a,x(a)) E

J.

5 x *( a ) 5

q(a,x(a))

x" = and

Suppose t h a t t h e r e e x i s t s a

x'(to)

> q ( t o , x ( t o ) ) . Define

30

15. EXISTENCE IN THE LARGE

Then,

i n some i n t e r v a l t o t h e l e f t of

to. This implies t h a t

V(t)

t increases and p o s i t i v e near to and hence on [a, toI. Thus we have V(a) > 0 o r equivalently x ' ( a ) > $ ( a , x ( a ) ) which i s a contradiction. Hence x ' ( t ) _< $ ( t J x ( t ) ) on J . S i m i l a r l y we can v e r i f y i s nonincreasing as

'p(tJx(t))

5 x'(t>

on J.

1.5 EXISTENCE I N THE LARGE We a r e now ready t o prove theorems on existence i n t h e large.

We begin with one of t h e b a s i c r e s u l t s i n t h i s d i r e c t i o n .

THEOREM 1.5.1.

Let

a,p

E

C(l)[J,R]

be, respectively,

lower and upper s o l u t i o n s of (1.3.1) on J

@(t) _< @ ( t ) on J.

Suppose f u r t h e r that

such t h a t f(t,x,x')

satisfies

Nagumo's condition on J any .(a)

(1.5.1)

5

c

5

r e l a t i v e t o t h e p a i r a,@. Then, f o r @ ( a ) , ~ ( b 5) d 5 B ( ~ ) J t h e BVP

x" = f ( t , x , x ' ) ,

has a s o l u t i o n x

E

x ( a ) = c,

c ( 2 ) [ J ~ R ] with

31

x ( b ) = d,

a ( t >_< x ( t > _< @ ( t )and

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

lx'(t)l

5N

on J,

where

depends only on a,@ and t h e

N

Nagumo's function h. By Theorem 1.4.1, t h e r e i s an N

Proof:

only on a,p,h with

a(t)

5

such t h a t

Ix'(t)l

x ( t ) _< @ ( t ) on J .

( c u ' ( t ) ( < cl,

(P'(t)(

<

5

on J

N

depending

f o r any solution

c > N so t h a t 1 Then, by Theorem 1.3.1, t h e

Choose a

on J.

c1

>0

BVP x" = F ( t , x , x ' ) , has a solution on J,

where

f(t,x,x')

x ( a ) = c,

x E C(2)[J,R]

a ( t ) 5 x ( t ) _< p ( t ) i s t h e modification f'unction of

F(t,x,x')

with respect t o

to

theorem t h e r e i s a

x ( b ) = d,

E Jo

such t h a t

a,p,

and

cl.

By t h e mean value

such t h a t

(b - a ) x ' ( t o ) = x ( b )

- x(a),

and as a r e s u l t using (1.4.3) it follows t h a t

Ix'(t,)l 5 h < N < cl. This implies t h a t t h e r e i s an i n t e r v a l containing toy where x ( t ) i s a solution of x" = f ( t , x , x ' ) . By Theorem

1.4.1, we have

Ix'(t)l

5

N

< c1 on t h i s i n t e r v a l .

However, I x ' ( t ) J < cl-

x ( t > i s a s o l u t i o n of

XI' = f ( t , x , x ' )

We conclude t h a t

i s a solution of (1.5.1) on J.

x(t)

as long as

The

proof i s complete.

COROLLARY 1.5.1.

Under t h e assumptions of Theorem 1.5.1,

any i n f i n i t e sequence of solutions of the relation

a(t)

,< x ( t ) _<

p(t)

XI'

on J,

= f(t,x,x')

obeying

has a uniformly con-

vergent subsequence converging t o a solution of

x'l = f ( t , x , x ' )

on J . A conclusion similar t o Theorem 1 . 5 . 1 may be drawn r e l a t i v e

t o t h e boundary value problem (1.3.1), Theorem 1.3.2,

EXercise 1.4.1,

(1.3.4), on the b a s i s of

and the proof of Theorem 1.5.1.

32

1.5. EXISTENCE IN THE LARGE

We state it in the following exercise. Let hypotheses (i) and (ii) of Lemma Show that for any A,B for which relations (1.3.6)

EXERCISE 1.5.1.

1.3.1 hold.

and (1.3.7) are valid, the BVP (1.3.1), (1.3.4) has a solution such that a(%) _< x(t) _< p(t) on J and x E C(*)[J,R] Ix'(t)l < N on J, where N > 0 depends o n l y on a,@,L. EXERCISE 1.5.2.

Let f E C [ J x R x R , R ] , f(t,x,x') be nondecreasing in x for each (t,x') and satisfy If(t,x,yl)f(t,x,y2)l5~ Iy1-y21 for (t,x> E J X R and yljy2 E R. Show that the BVP (1.3.1), (1.3.4) has a solution. If f is strictly increasing in x, then show that the solution is unique. Hint: -

Set M

max f(t,O,O), m = min f(t,O,O) and J J let p(t) > - 0, a(t) _< 0 be solutions of x" + L l x ' I - m = 0 , XI'- Llx'l - M = 0 , respectively, subject to (1.3.6), (1.3.7) which may be computed explicitly. Then, using monotony of f, show a,@ are lower, upper solutions. For uniqueness, use =

Theorem 1.2.4. 2

1.5.3. Verify that p(t) = t, a(t) = t 1 - 3t +2 are upper, lower solutions for the problem x'' = - 1x1' + t, x(1) = 0, x'(2) = 1, on J = [1,2]. Draw conclusions on the applicability of Exercises 1.5.1, 1.5.2 to this problem. Discuss the merits. EXERCISE

The foregoing discussion concerning the problem (1.3.1),

(1.3.4) imposes more restrictions on the function f compared This is apparently due to the techniques to the B W (1.5.1). involved in the process rather than the need, because the result that we present next involves nonlinear boundary conditions more general than (1.3.4) and yet the assumptions on f are just 33

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

those required f o r t h e BW (1.5.1). We s h a l l f i r s t prove t h e following.

THEOREM 1.5.2. (a) a,@

E

Assume t h a t

C(l)[J,R],

s o l u t i o n s of (1.3.1) on

respectively, a r e lower and upper such t h a t

J

a(t) 5 B(t)

on J

and

a(a) < B(a); (b)

s a t i s f i e s Nagumo's condition on

f(t,x,x')

respect t o t h e p a i r (c) y

g

E

f o r each

(1.5.2)

a(b) J

_<

d

g(x,y)

i s nondecreasing i n

and

g(a(a),a'(a))

2

g(B(a),B'(a))

0,

It i s enough t o show t h a t given

Proof:

with

a,B;

C[[a(a),B(a)] xR,R],

x,

J

_< f3(b),

there i s a solution

x(t,E)

5

0.

E

>

0

and

of (1.3.1)

on

that fulfills x(b,E> = d,

and

a ( t ) 5 x(t,E)

that that such

on

<E

J.

_< d _< B(b), l e t T(c) denote t h e s e t (1.5.1) such t h a t a ( a ) _ < c 0 , t h a t f o r every a ( a ) _< c 5 p ( a ) and x ( t ) E ~ ( c ) , we have For a f i x e d

of s o l u t i o n s and

5 B(t)

Ig(X(a,E), x'(a,E))I

x(t)

a(b)

t o t h e BW

Ig(x(a),x'

(4)I 2 E O . 34

1.5. EXISTENCE IN THE LARGE

Define the set S by

and let co = sup[x(a) = c: x(t) E S]. We notice that x(t) E T(cy(a)) implies a'(a) _< x ' ( a ) which by (1.5.2) yields g(x(a>,xl(a>) E ~ . Similarly, x(t> E .rr(@(a)) implies

>

5 -EO. This observation x'(a) _< @'(a) so that g(x(a),x'(a)) proves that co < p(a). Let yo(t) be a solution of (1.3.1) which is obtained as a uniform limit of members of S so that (1-5.3) yO(a> =c0,

Yo@) = d ,

and g(Yo(a>, Y;)(a))

2

€0.

For n 2 N, Now let N 2 1 be such that co + (1/N) 5 @(a). satisfying, in addition, the inlet yn(t) E T(C~ f (l/n))

>

equality yn(t) yo(t) on J. This is clearly possible since yo(t) may be treated as a lower solution of (1.3.1). Then a subsequence of yn(t) converges to a solution xo(t) obeying xo(t) yo(t) on J. By definition of co,

>

E T ( C ~ )

we have

5 -cO and consequent- g(xo(a>, x;(a)) _< g(yn(a), y;(a)) Since x;(a) 2 y;)(a) and xo(a) = yo(a), this leads to g(y,(a), y;)(a)) 5 - E O which is a contradiction in view of (1.5.3).

-E~.

The proof is complete.

Employing similar arguments as in the proof of Theorem 1.5.2, which were based on a consequence of Theorem 1.5.1, namely Corollary 1.5.1, we may prove the following theorem. THEOREM 1.5.3.

Assume that

respectively, are lower and upper C(l)[J,R], solutions of (1.3.1) on J such that (a) a,@

E

a(t)

5

@(t)

on J and a(b) < @(b);

(b) f(t,x,x') satisfies Nagumo's condition on J relative to the pair a,@; 35

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(c) h

C[[a(b),B(b)]

E

xR,R],

y for each x and h(a(b),

=

f(t,x,x'),

5

a'(b))

Then, for any .(a) 5 c 5 @(a) x C E C(2)[J,R] of the BVP x"

h(x,y)

2

h(B(b),B'(b))

0.

there is a solution

x(a) = c,

which satisfies a(t)

0,

is nondecreasing in

5 xc(t) 5

h(x(b),x'(b))

B(t)

= 0

on J.

Combining the proofs of Theorems 1.5.2 and 1.5.3, we may obtain our main result relative to nonlinear boundary conditions. THEOREM

1.5.4. Assume that

(a) a,@

E

C(l)[J,R],

respectively, are lower and upper

solutions of (1.3.1) on J .(a>

such that Q(t)

5 B(t)

on J and

4 b ) < Bb); (b) f(t,x,x') satisfies Nagumo's condition on J with

< B(a),

respect to the pair a,B; E C[[a(a),B(a)l xR,R], h E C[[o(b),B(b)] xR,R], (c) g(x,y), h(x,y) are nondecreasing in y for each x and

g(a(a),al(a)) h(a(b),a'(b))

2

0,

5

0,

Then there is a solution x XI' = f(t,x,x'),

g(B(a),B'(a)) h(B(b),B'(b)) E

which satisfies a(t)

= 0,

5 x(t) _<

@(t)

Proof: For each a(b) 5 d set of solutions of the BVP

x"

=

f(t,x,x'),

5

2

0-

h(x(b),

x'(b))

= 0,

on J.

@(b),

g(x(a>,x'(a))

0,

of the BVP

C(*)[J,R]

g(x(a),x'(a))

5

let T ( d ) denote the = 0,

x(b) = d,

which satisfy a(t) 5 x(t) 5 B(t) on J. By Theorem 1.5.2, is nonempty for all it is clear that the set T(d) 36

15. EXISTENCE IN THE LARGE

which contradicts the supposition that the theorem is not Now use y(t) in place of a(t) true. Hence y(a) < @(a). in Theorem 1.5.2 to conclude that the set D = [x(t): x(t)

E

T(d),

do

< d 5 B(b),

y(t)

5 x(t)

.< @(t>

on

JI

in nonempty. Moreover, x(t) E n(d) implies that h(x(b), x'(b)) > 0 . However, inf[x(b) = d: x(t) E D] = do, which leads to a contradiction by a convergence argument and the fact that h(x,y) is nondecreasing in y. Hence the proof is complete. On the basis of Theorem 1.4.3, it is possible to relax the condition of the basic Theorem 1.5.1 which is the content of the next result. THEOREM 1.5.5. Suppose that assumptions (i) and (ii) of Theorem 1.4.3 hold. Let a,@ be lower and upper solutions of (1.3.1), respectively, on J, such that a(&) = @(a) and

cp(t,y(t))

5 ~'(t)I +(t,r(t>),

for y(t) 37

=

a(t),B(t),

t

E

J.

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Then, f o r any

5

a(b)

d

_< B(b) and u(a)

BVP (1.5.1) has a solution

x E C(2)[J,R]

(t,x(t),x'(t>) where

E

n,

t

= c = B(a),

the

such t h a t E J,

il i s t h e s e t given by

R = [(t,x,x'):

Proof:

(t,x>

E

(0

and

cp(t,x)

Define a modified function

5 x' 5

Jr(t,x)l.

F(t,x,x')

on

cuxR

by f(t,x,dt,x)) f(t,x,+(t,x)) and extend

F(t,x,x')

to

if x '

< cp(t,x>,

if

(t,x,x')

if

XI

>

E

n,

q(t,x),

J x R x R by s e t t i n g

F(t,a(t),x')

if

x < a(t),

F(t,B(t),x')

if

x > @(t).

F(t,x,x') = Observe t h a t

i s bounded on J x R x R

F(t,x,x')

s a t i s f i e s Nagumo's condition.

a',@'), w,B,

assumptions on

and therefore

Furthermore (because of t h e a r e a l s o lower and upper solutions

relative t o

Consequently, by Theorem 1.5.1 t h e r e exists a solution x E C(2)[J,R]

t o t h e modified BVP

x" = F(t,x,x'), such t h a t and

a(b)

a(t)

5

d

x ( a ) = c,

x(b) = d,

5 x(t) 5

_< B(b).

B(t), t E J f o r .(a) = c = B(a) We now apply Theorem 1.4.3 t o conclude

i s a c t u a l l y a s o l u t i o n of t h e BVP (1.5.1) s a t i s f y i n g ( t , x ( t ) , x ' ( t ) ) E R, t E J. Hence t h e proof i s

that this

x(t)

complete.

38

1.6. LYAPUNOV-LIKE FUNCTIONS

1.6.

LYAPUNOV-LIKE FUNCTIONS We wish t o employ, i n t h i s section, Wapunov-like functions

and the theory of d i f f e r e n t i a l i n e q u a l i t i e s t o e s t a b l i s h existence i n t h e large f o r t h e BVP (1.5.1).

+

We define t h e generalized derivative

x" = f ( t , x , x t )

the d i f f e r e n t i a l equation Dt,( t

J

X, X

V

Let

C[JxRxR,R].

E

D Vf(tJxJx') relative t o

by

')

- V(t,x,x')].

lim sup 1 [V(t + h , x + h x ' , x ' + h f ( t , x , x ' ) ) h+O+ h We s h a l l often use s e v e r a l f'unctions ViJ

i = 1,2,.

..,n,

which

we c a l l I(fapunov-like f'unctions and f o r n o t a t i o n a l consistency we write Vi

of

+

D V. ( t , x , x t )

t o denote t h e generalized derivative

If

with respect t o t h e d i f f e r e n t i a l equation

(1.6.1)

XI1 =

THEOREM 1.6.1.

(i) F(t,x,x') the t r i p l e (ii)

Assume t h a t

a , p c C(l)[J,R]

such t h a t

i s t h e modified f'unction of

a,p,c,

a(t)

-< p ( t )

f(t,x,x')

on

J

and

relative t o

as given i n t h e Definition 1.3.1;

t h e r e e x i s t two Wapunov functions

wi

x [x: x ,< a ( t ) ] xR, D2

= J

i = 1 J 2 J . where

B(t)]xR,

f(t,X,X').

D1

= J

E

C[D~,R],

x [x: x >

W ( t , x , x ' ) = 0 i f x = a ( t ) , Wl(t,x,xl)>O 1 x < a ( t ) , W2(t,x,xt) = 0 i f x = B(t), W2(t,x,x1) > O i f

if

such t h a t

x > B(t); (iii)

for

i = 1,2, g

locally Lipschitzian i n

+

D WP(t,x,xt)

+

D W2F(t,x,xt)

i

E

+

C[JxR , R ] , Wi(t,x,xl)

is

( x , x t ) and

_> gl(t,Wl(t,x,xl)) g2(t,W2(t,x,x1))

39

i n t h e i n t e r i o r of

D1j

i n t h e i n t e r i o r of

D2;

1.

to

E

METHODS INVOLVING DIFFERENTIAL INEQUALITIES

( i v ) the only solution of J , i s r ( t ) p 0 on J .

5

Then, if .(a) (1.6.2)

c

5

has a solution x

d

5

r ( t o ) = 0,

the BVP

B(b),

x ( a ) = c,

C(2)[J,R]

E

5

B(a), a ( b )

x" = F ( t , x , x ' ) ,

r ' = gi(t,r),

x(b) = d,

satisfying

a(t)

-< x ( t ) 5 p ( t )

on J . Proof: x

E

By Corollary 1.1.1, the BVP (1.6.2) has a solution

C(2)[J,R].

on J .

It s u f f i c e s t o show t h a t

We w i l l only prove t h a t

proof f o r

x(t)

5

[tl,t21C (a,b)

E

such t h a t

< a ( t ) for t

E

a(t)

5 x(t)

-

< x(t) < @(t) on J since the

i s e s s e n t i a l l y the same.

< a ( t ) f o r some t

x(t) x(t)

p(t)

a(t)

(a,b).

Then t h e r e e x i s t s an i n t e r v a l

x ( t l ) = a(t,),

(tl,t2).

Assume t h a t

x(t2) = a ( t 2 ) ,

and

We thus have by (ii)

From ( i i i ) and the theory of d i f f e r e n t i a l i n e q u a l i t i e s , we obtain (1.6.4) where

Wl(t,x(t),x'(t)) r(t,t2,ro)

r ( t2 ) = r 0 clear that

5

r(t,t2,ro),

a

5

t

i s the l e f t maximal solution of

5 t2, r ' = gl(t,r),

~y ( i v ) and (1.6.3), it i s 1( t2, x ( t 2 ) , x t ( t 2 ) ) . we r ( t , t 2 , 0 ) 1 0 . However, f o r any t E (tl,t2),

= W

obtain, because of ( i i ) and (1.6.4), the contradiction 0

< W1(t,x(t),x'(t)) 5

0.

The proof i s therefore complete. We notice t h a t t h i s theorem generalizes Theorem 1 . 3 . 1 . In case

a(t), p(t)

respectively, on J,

a r e lower and upper solutions of (1.6.2),

it is enough t o choose W l ( t , x , x r ) = a ( t ) - x 40

1.6. LYAPUNOV-LIKE FUNCTIONS

It i s easy t o check that

and W2(t,x,x') = x - p(t).

Wl

s a t i s f i e s a l l the assumptions of Theorem 1.6.1 w i t h g1 = 0 . The only condition t h a t requires a l i t t l e explanation i s ( i i i ) , + For this, namely D WD(t,x,x') 2 0 i n the interior o f D

1' does not a t t a i n a it is sufficient t o prove that Wl(t,x,x') localmaxhum i n the interior of Arguing as i n the proof D1. of Theorem 1.3.1, we see that it i s necessary t h a t a ( t l ) = x ( t ) 1 a t some tl E (a,b) i n order for W1 t o a t t a i n a local maxjmum.

However, for t = t we obtain DWb(tl,x(tl), x ' ( t l ) ) > 0 as 1' i n the proof o f Theorem 1.3.1 which i s impossible a t a local maximum of W1. A similar reasoning holds for W2 and thus the conditions of Theorem 1.6.1 are verified. The next theorem offers a general s e t of conditions t o ensure that / x ' ( t ) I is bounded.

-

Assume that a,p E C(l)[J,R] and a ( t ) < p(t) on J . Define the s e t s El = J x [x: a ( t ) 5 x 5 p ( t ) ] x [x': X ' ,> 01 and E2 = J X [x: a ( t ) ,< x @ ( t ) ]X [x'; X I 5 01.

THEOREM 1.6.2.

Suppose that there exist Iyapunov f b c t i o n s Vi, i = 1,2,3,4, that are locally Lipschitzian i n (x,xl) such that ( i ) Vi E CIE1,R] and Vi(t,x,x') + m as uniformly on J x [x: a ( t ) < x 5 B(t)], i = 1,2; and ( i i ) D+Vlf(t,x,xl) 5 gl(t,Vl(t,x,x'))

-

XI

+m

-+

D V2f(t,x,x') 2 g2(t,V2(t,x,x')) on El; ( i i i ) Vi c C[E2,R] and Vi(t,x,x') + m as x' +-uniformly on J X [x: a ( t ) x 5 B(t)], i = 3,4; + (iv) on E2, D V3f(t,x,xt) -c g3 (t,V3(t,x,x')) and -i.

D V4f(t,X,X-) ,> gq(t,Vq(t,x,x')); (v) Vi(t,x,x') 5 Jri(lxII), i = 1,2,3,4, for r > 0 are continuous f b c t i o n s . Suppose f i r t h e r that

41

where

Jli(r)>O

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(Vi)

gi

C[JXR,R], i = 1’2’3’4

f

and f o r

to f J, ro

E

R,

a l l solutions of

e x i s t on rt = g i ( t J r ) , r ( t o ) = roJ i = 1 , 3 [ t O , b l J and a l l solutions of rt = g i ( t J r ) ’ r ( t ) = r

i = 2,4

e x i s t on

Then there e x i s t s an N with

a(t)

Define f o r

>0

-< x ( t ) -< @ ( t )

t

0

[aJto].

0’

such t h a t every solution of

on

J

satisfies

Ixt(t)l

-<

N

(1.6.1) on

J.

E J

and L.1 =

max

to E J

lri(tJtoJqi(A))

IJ

i =

2’4,

where ri(tJtOJjri(h)), i = 1’3 (i = 2’4) i s the r i g h t ( l e f t ) maximal solution of rt = g i ( t J r ) ’ r ( t o ) = qi(A). Then cond i t i o n ( v i ) guarantees t h a t t h e L.’s a r e f i n i t e . By ( i ) and ( i i i ) , there e x i s t

(1.6.5)

Ni

>0

1

i = 1,2

3 4 such t h a t

> Lij

i = 1’2’

v.1(tjxj-Ni) > Lij

i = 3’4.

Vi(tjXjNi)

and

kt N >maX[Nijh]y i = 1J2j3y4. We Claim t h a t IX’(t)l 5 N on J where x ( t ) i s any solution of (1.6.1) with a ( t ) 5 x(t)

5

@ ( t ) on J .

There e x i s t s a

to E (a,b)

42

such t h a t

x(b)

- x ( a ) = x ’ (tO)(b-a)

1.6. LYAPUNOV-LIKE FUNCTIONS

and therefore it follows t h a t Ix' ( t o ) I 5 h < N. There are four cases t o be considered depending upon whether there e x i s t s

t2 such t h a t x ' ( t 2 ) = N or x ' ( t 2 ) = -N and whether t2 > to o r t2 < to. Assume, f o r example, x ' ( t 2 ) = N and a

t 2 > to. Then there e x i s t s a tl with that

xl(t,) = A

h < xl(t)

and

t

< N for t

(v) we obtain Vl(tl>x(tl)Jxl(tl)) arrive a t the estimate

5

< t 1 < t2 such

0 -

E

(t,,t,).

From

$1(~) and from ( i i ) we

using the theory of d i f f e r e n t i a l inequalities. Relations (1.6.5) and (1.6.6) lead us t o the contradiction L1 < v1(t2JX(t2)JX' ( t 2 ) )

because of the definitions of

5

L1

L1 and N.

On the basis of the preceding argument, we can arrive a t a similar contradiction i n the remaining three cases. the d e t a i l s .

We omit

The proof i s therefore complete.

The r e s u l t we have j u s t proved offers a general s e t of sufficient conditions which imply that

Ixl ( t ) I

i s bounded

and thus relaxes the more stringent Nagumols condition assumed i n Theorem 1.4.1.

In f a c t , the intent of the following

exercise i s t o c l a r i f y t h i s advantage f u r t h e r . EXERCISE 1.6.1.

Let

hi

E

C[R+,(O,m)], f

E

C[JxRxR,R],

and

Suppose t h a t

t

E

J

and

aJf3E C[J,R]

a(t)

5x5

with w ( t ) < B(t)

@ ( t ) J assume that

43

on J.

For

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

-

-h2(xf) c f ( t , x , x ' ) -h (-XI)

3

5 f ( t , x , x ' ) 5 hq(-xl)

Show t h a t for any solution x a ( t ) 5 x ( t ) 5 B(t) on J, only on cu,p,hi such t h a t Hint: -

hl(xl)

E

C(2)[J,R]

if

x'

if

x'

-> 0, -< 0 .

of (1.6.1) w i t h

there e x i s t s an N > 0 I x l ( t ) l L N on J .

depending

Under the assumptions show one can construct four

Qfapunov functions satisfying the conditions of Theorem 1.6.2. For example s e t V1 = exp[t - x + ./:' s ds/hl(s)] on El show the assumptions are verified w i t h g l ( t , r ) = r and and V4. similarly construct V2, V

and

3'

we may ccanbine Theorems 1.6.1 and 1.6.2 t o obtain the existence of solutions (1.5.1). "he proof i s similar t o Theorem1.5.1. Wemerely statethefollawingtheoremwhichisagener~izationofTheorem1.5.1.

THEOREM 1.6.3.

-

a,@ E C(l)[J,R] w i t h a ( t ) c B(t) Assume t h a t there e x i s t two l$fapunov f'unctions W1, W2 Let

on J . s a t i s Q i n g the hypotheses of Theorem 1.6.1. Suppose further t h a t there exist four Qrapunov functions Vi, i = 1,2,3,4 such t h a t they obey the hypotheses of Theorem 1.6.2. Then f o r any .(a) c c c p(a), u(b) 5 d 5 p(b) t h e BVP (1.5.1) has a solution x E C(*)[J,R] with a ( t ) c x ( t ) C p ( t ) and Ix'(t)l S N on J .

- -

-

-

1.7 EXISTENCE ON INFINITE INTERVALS On the basis of Theorems 1.4.1 and 1.5.1 it i s possible t o obtain the existence of solutions on i n f i n i t e intervals.

THEOREM 1.7.1. Assume t h a t f o r each b > a , f ( t , x , x ' ) s a t i s f i e s Nagumofs condition on [a,b] r e l a t i v e t o the p a i r a,B E C(l)[[a,m),R] with u ( t ) 5 B(t) on [a,-). Suppose 44

1.7. EXISTENCE ON INFINITE INTERVALS

also that a , @ are lower and upper solutions of (1.6.1) on [a,m), respectively. Then f o r any a ( a ) 5 c 5 @(a) the B W (1-7.1)

xtt = f(t,x,x'),

he8 a solution

x

on

E

x(a) = c,

C(2)[[a,m),R]

-

such t h a t a ( t ) c x ( t ) < p(t)

[a,m). Proof: -

By Theorem 1.5.1, it follows that f o r each n 1 1 there i s a solution xn E C(2)[[a,a+n],R] such t h a t xn(a) = c,

xn(a+n) = p ( a + n ) ,

and a ( t ) 5 x n ( t )

5 p ( t ) on [ a , a + n l .

-

Furthermore, there i s an Nn > 0 such t h a t I x t ( t ) l < Nn on [ a , a + n l f o r any solution satisfying a ( t ) 5 x ( t ) 5 p ( t ) on > 1, x,(t) is a solution on [a,a+n]. Thus f o r any fixed n [a,a+nI verifying a ( t ) 5 x,(t) 5 p ( t ) and Ix;(t)l 5 N~ on [ a , a + n ] f o r a l l m > n. Consequently, f o r m i n the sequences {xm(t)], {x;(t)] are both uniformly bounded and equicontinuous on [a, a + n ] Then, employing the standard diagonalization arguments, we obtain a subsequence which converges uniformly on a l l compact subintervals of [ a , ~ ) t o a solution x ( t ) . The desired solution of (1.7.1) is precisely this x ( t ) . Hence the proof is complete.

.

Essentially a similar proof may be given t o the following, s a t i s f i e s Nagumo's condition on [-a,a] f o r each a > 0 w i t h respect t o the pair @,p E C(l)[R,RI w i t h a ( t ) 5 p ( t ) on R. Suppose also t h a t (Y,B are lower and upper solutions of (1.6.1) on R . Then there i s a solution of (1.6.1) on R such t h a t a ( % ) 5 x ( t ) 5 B(t) on R. !MEOREM 1.7.2.

Assume t h a t

f(t,x,xt)

We shall merely s t a t e below a r e s u l t analogous t o Theorem 1.7.1 i n terms of Qfapunov-like functions.

45

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

THEOREM

on

[a,m)

1.7.3. L e t a,@

and l e t

f

f

c C(')[[a,-),R]

C[[a,-)xRxR,R].

with a ( t ) 5 p ( t ) Assume there e x i s t

two Wapunov functions Wl(t,x,xl), W2(t,x,x1) obeying the hypotheses of Theorem 1.6.1 where we now replace

[a, b 1 with

[a,m).

Assume also t h a t there e x i s t two Iiyapunov functions V2, V4 s a t i s w i n g the corresponding properties of !Theorem 1.6.2 w i t h [a,bl replaced by [a,-). Wre precisely suppose that ( i ) V2(t,x,x') subsets of

-

[a,m) x [x: a ( t )

+

(ii) D V2f(t,x,xl)

a ( t ) < x 5 B ( t ) and x' ( i i i ) V4(t,x,xt) 4 subsets of

as

-t-

uniformly on compact

+Q)

5 x 5 @(t)];

3 g2(t,v2(t,x,x1)) for t

2

E

[a,-),

0;

m

[arm) x [x: a ( t )

+

XI

as

x'

5x5

uniformly on compact

--f-c=~

p(t)l;

( i v ) D V4f(t>x,x1)2 g4(t,v4(t9x,xt)) f o r t E [a,m), a(t) 5 x B(t) and x ' 5 0; (v) there e x i s t qi, i = 2,4 where qi e C([a,m) xR,R] and Qi(t,u) i s increasing i n t for each u such t h a t f o r each

2 a, Vi(t,x,x') e qi(T, Ix' I ) 5x5 T

for

t

E

[a,T]

and

(vi) gi E C[[a,-) XR,R], i = 2,4 such t h a t f o r each t 0 e [a,-) and ro 2 0 a l l solutions of r' = gi(t,r), r ( t o ) = ro e x i s t on [a,tol. !Then for any

c

satisf'ying

a(.)

6 c 5 B(a) there e x i s t s

-

a solution of (1.7.1) such that a ( t ) c x ( t )

5 B(t)

on

[a,-).

Clearly a theorem analogous t o Theorem 1.7.2 may be formulated, which we leave as an exercise.

1.8 SUPER- AND SUBFUNCTIONS Here we define subf'unctions and superfunctions r e l a t i v e t o the solutions of XI' = f ( t , x , x ' ) and discuss necessary and 46

1.8. SUPER- AND SUBFUNCTIONS

sufficient conditions I o r such functions t o be, respectively, lower and upper solutions.

To avoid repetition, most of the

results w i l l be stated only i n terms of lower solutions and When it becomes necessary l a t e r t o r e f e r t o a

subfunctions.

result concerning superfunctions and upper solutions, we s h a l l simply r e f e r t o the subfunction statement of the r e s u l t .

DEFINITION 1.8.1.

A function

function r e l a t i v e t o solutions of val J

[tl,t2]

i f f o r any

C

Cp(t)

is said t o be a

x" = f ( t , x , x ' )

2-

on an inter-

and f o r any solution

J

.)> cp(t.), i = 1,2 implies x ( t ) 2 cp(t) C(2) [[tl,t21,R] , x (t 1 1 on [t t 1. A superfunction may be defined similarly by re1 ' 2 versing the respective inequalities.

x

E

We have immediately the following r e s u l t . THEOREN

1.8.1.

Assume t h a t

cp c C[J,R] I I C(l)[J0,R]

is a

subfunction on J w i t h respect t o the solution of

(1.8.1)

XI1

where f E C[J X R xR,R]. (1.8.1) on J . Proof:

Let

Then

to 6 Jo

sufficiently small.

= f(t,X,X') Cp

and h > 0, k > 0, h

-

E

C(2)[[t0-k,t0+h],R].

subfunction, we readily obtain Cp'

-

be

x ( t O+ h ) = q ( t o + h ) ,

x ( t O k) = g ( t 0 k),

has a solution x

+k >0

Then by Theorem 1.1.2 t h e BVP

-

X" = f(t,X,X'),

i s a lower solution of

-

(to+ h ) Cp' ( t o k) h+k

->

X' (to + h )

X'

cp

is a

-

(to k)

h+k =

for some t o- k < s < to + h.

-

Since

x'l(s) = f ( s , x ( s ) , x ' ( s ) )

!the continuity of 47

f

together

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

with "heorem 1.1.2, implies that f ( s , x ( s ) , x t ( s ) ) --3 f ( t O , x ( t O ) , x t ( t O ) )

as h + k

+ 0.

Thus

and, i n particular,

from which it follows t h a t (1.8.1) on J.

i s a lower solution r e l a t i v e t o

cp

Assuming only the continuity of f , it w i l l not be possible i n general t o show t h a t a lower function is a subf'unction. Since a solution i s a lower solution, if lower solutions are subf'unctions, then solutions are subf'unctions From the definition of subfunctions it would then follow t h a t , if a BVP on an i n t e r v a l has a solution, thet solution is unique. Hence stronger assmptions other than continuity of f are required t o conclude that a lower solution i s a subfunction. A s e t of sufficient conditions i s given i n the next theorem.

.

THEOREM 1.8.2.

Assume t h a t

( i ) f E C[JxRxR,R] and i n x f o r each (t,g); ( i i ) either f satisfies a i n y on each compact subset of i n i t i a l value problems f o r (1.8.1) Then a lower solution on J t o solutions of (1.8.1).

f(t,x,y)

i s nondecreasing

one-sided Lipschitz condition J x R x R o r solutions of are unique.

is a subfunction on J r e l a t i v e

The proof i s a direct consequence of Theorems 1.2.2 and 48

1.8. SUPER- AND SUBFUNCTIONS

1.2.3.

I n t h e preceding r e s u l t conditions a r e imposed on f ( t , x , y ) which a r e s u f f i c i e n t t o imply t h a t lower solutions a r e subf'unctions and, therefore, t h a t solutions of BVP's, when they

I n t h e next r e s u l t we take t h e uniqueness

exist, a r e unique.

of BVP's as one of t h e hypotheses. THEOREM 1.8.3. Assume t h a t each i n i t i a l value problem f o r (1.8.1) has a s o l u t i o n which extends throughout [a,b]. Suppose

further t h a t solutions of boundary value problems X"

= f(t,X,X'),

x ( t l ) = xl,

~ ( 6 =~x2, )

[tl,t2] C J,

when they e x i s t , a r e unique.

Then, i f

i s a lower solution on

i s a subfunction on I.

Proof:

Assume t h a t

there i s an i n t e r v a l such t h a t (c,d).

CY E

is a

x,(c)

C(l)[[c,d],R],

>0

cy

[c,d]

and

i s not a subfunction on C

I

and solution

F(t,x,x')

on

C(l)[I,R]

cy E

I.

Then

xo E C(2)[[c,d],R]

and x,(t) [c,d] X R x R

< a(t)

on

by

i s continuous on [c,d] X R X R and it follows from Theorem 1.1.2 t h a t there

F(t,x,x') 6

cy

= cy(c), xo(d) = cy(d),

We now define

Since

I,

IC J

such t h a t

[tl,t2]

C

[c,d]

and

t2-tl 5

6

implies the BVP X"

= F(t,x,x'),

has a solution

x

x(t,)

= Q(tl>,

E C(2)[[tl,t2],R].

x ( t 2 > = a(t2),

Using t h e f a c t t h a t

a ( t ) 5 x ( t ) on with t h e same type of argument as used i n t h e proof

i s a lower solution, we can show t h a t [tl,t2]

Ly

of Theorem

1.3.1.

As a r e s u l t , f o r 49

[t1,t2] C [c,d]

and

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

t2- tl

5 XI1

t h e BVP

6, =

f(t,X,X'),

has a s o l u t i o n x

x(tl) = a(tl), C(2)[[t,,t,],R]

E

x ( t 2 ) = a(t,),

with

a(t) < x(t)

on

[tl,t2]. Thus a i s a subfunction " i n t h e small." Clearly d c > 6; f o r otherwise t h e r e would be a s o l u t i o n

-

x(t)

with

[c,d].

x(c) = a ( c ) , x(d) = a(d),

an% x ( t )

-> a ( t )

This s o l u t i o n would be d i s t i n c t from x ( t ) 0

on

contra-

d i e t i n g t h e assumption concerning t h e uniqueness of s o l u t i o n s of BVP's.

Now f o r each p o s i t i v e i n t e g e r

n,

let

P(n)

t h e proposition t h a t t h e r e e x i s t s an i n t e r v a l [cn,dn] with 0 < dn - c < d - c - (n 1)s and a s o l u t i o n n-

-

x n on

E

C(2)[[cn,dn],R]

such t h a t

d i s t i n c t s o l u t i o n with boundary values

X"

C

[c,d]

x (c ) = a ( c n ) , xn(d ) = a ( t ) n n n i s t r u e with [cl,dl] = [ c , d ]

( c , d ). Evidently P(1) n n and x 1( t ) = x o ( t ) . Assume P(k) i s t r u e . Then otherwise we would o b t a i n a c o n t r a d i c t i o n o f x , ( t )

z,(t)

be

a(ck)

4,- Ck >

6,

being t h e

and a ( % ) .

Let

be t h e s o l u t i o n of t h e BVP =

f(t,X,X'),

x(c,)

=

X ( Ck + 6) = a ( c k + 6 ) .

CU(C,),

Since each i n i t i a l value problem has a s o l u t i o n extending throughout t h a t z,(t)

J,

i s not t r u e .

t h e r e i s a s o l u t i o n z 2 ( t ) on [ck,%I such z,(t) on [ c k , c k + 6 ] . Suppose t h a t P ( k + l ) Then we must have

Also we must have

z2(%)

z2(t)

> "(4,). If

2 a ( t ) on [ c k + 6 , 4 , ] . 4, - ck - 6 5 6, t h e

BVP

E C ( 2 ) [ [ ~ k + 6 , 4 , ] , R ] with a ( t ) < z ( t ) . z3 - 3 Then, s i n c e z ( ) < z2(4,) and s o l u t i o n s of BVPls a r e unique 3 % t E cCk+6,%1. a(t) 5 z 3 w 5 z , ~ ,

has a s o l u t i o n

50

1.8 SUPER- AND SUBFUNCTIONS

This implies z (Ck+6)

3

= z2(Ck+6)

Consequently,

u(t),

and

Z'(C

3

k

+ S ) = Z'(C +6).

2

k

defined by

i s of c l a s s C(2)[[ck,%],R]

and i s a s o l u t i o n on

[ck,%]

However, u ( t ) # \ ( t ) with u ( ck ) = %(ck), u($) = . on rek,%] and t h i s c o n t r a d i c t s t h e uniqueness of s o l u t i o n s of BVP's.

We conclude t h a t

% - ck >

This being t h e case,

6.

the BVP X"

= f(t,X,x'),

has a s o l u t i o n

x(ck+ S ) = a(ck+ S),

z4

E

X(C +26) k

C(2)[[c + 6,ck+26],R]

k

+ 6) 4 k

= z;(ck+ 6).

v(t) =

!Phis s o l u t i o n P(k+1)

on

v(t)

z4(ck + 6 ) = z2(ck + 6) Hence

z,(t)

on

[ck,ck+61,

z4(t)

on

(ck+ 6,ck+281,

has an extension

z (t)

5

a(ck+28),

with

This again assures us t h a t z'(c

=

and

on J .

Since

i s assumed t o be f a l s e , we must have

[ck+2S,%]

and

z,(%)

> a(\).

z5(t) 2 a ( t ) I n t h i s way t h e fore-

going arguments can be repeated and t h e assumption t h a t 51

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

P(k+l)

i s f a l s e permits us t o work our way across the

interval

[ckJ%] by subintervals of length

6 u n t i l we

> a ( t > on a(%) = \(%).

obtain a solution w c c(*) [ [ c k J % l , ~ 1 with w ( t >

[C,J%]J w(ck) = a ( ck = s ( c k ) and w(%) = xJt) on [c,J%] which again i s a contradiction Then w(t)

+

t o uniqueness of solutions of BVP’s. We therefore conclude t h a t , i f i s true.

P(k+l)

leads t o the contradiction 0 Hence a

P(k)

i s true, then

i s true f o r a l l n

Thus P(n)

i s a subfunction on

2

1, which

< d - c - (n - 1)s f o r a l l n > I

1.

and the proof i s complete.

Show t h a t i n Theorem 1.8.3 the hypothesis

EXERCISE 1.8.1.

t h a t each i n i t i a l value problem f o r (1.8.1) has a solution which extends throughout

can be replaced by Na@;umo’s condition.

J

1.9 PROPERTIES OF SUBFUNCTIONS Before proceeding t o outline the Perron’s method, it w i l l be necessary t o make a more detailed examination of the propert i e s of subt h i s section.

and superf’unctions.

This w i l l be undertaken i n

Again most r e s u l t s w i l l be stated i n terms of

subf’unctions and the obvious analogs f o r superf’unctions w i l l be omitted. IMEOREM I C J,

then

1.9.1. cp

If

cp

has r i g h t and l e f t limits i n the extended

r e a l s a t each point i n

Io and has appropriate one-sided

limits a t the end points of

Let

i s a subfunction on an i n t e r v a l

I.

Proof: Clearly it i s sufficient t o consider one case. to E Io and suppose t h a t cp(t-) 0 = limt+t_, cp(t) does

not e x i s t i n the extended r e a l s . numbers a,@ such t h a t

52

Then there e x i s t r e a l

1.9. PROPERTIES OF SUBFUNCTIONS

cp(t) < a < p < l i m sup cp(t)

lim inf t4t; It,},

Let

{s,]

t4;

be s t r i c t l y increasing sequences i n

I

t n < sn < tn+l f o r n 2 1, l i m t n = lim sn = to,

that

iim cp(tn) Taking

iim sup cp(t) t-tt;

=

t ( B -a),

E =

6> 0

is a

x" = f(t,X,X'),

has a solution x $(a+B)I

it follows from Theorem 1.1.2 t h a t there

<

[t,,t2]

C

[tl,tO]

with

the BVP

t2 tl

(1.9.1)

lim cp(sn) = lim inf cp(t). t+t;

and

such t h a t f o r any

- -< 6

such

E

E

on

x ( t 1) = x ( t 2 ) = $ ( a + p ) ,

C ( ~ ) [ [ ~ , , ~ ~ I satisfying ,R]

[tl,t2].

Let

Ix(t)

-

n be a fixed positive integer

-

sufficiently large so t h a t

sn+l sn -< 6, rp(sn) < a, q ~ ( s ~ <+a~,) and cp(tn+l) > p. This implies t h a t there i s a solution of (1.9.1) with [ t , t 1 = [S , S 1 and 1 2 n n +l Since ]. cp i s a subfunction lx(t) -$(a + B) I < E on [ S ~ , S ~ + ~ on 1 and cp(sn) < X(Sn)' cp(sn+l) < X(sn+l), it must be the case t h a t

cp(tn+l)

5 x(tn+l).

X(tn+l> <

3. + B )

However, + E

,

a contradiction from which we conclude extended r e a l s .

cp(t',)

e x i s t s i n the

The proof i s complete.

COROLIARY 1.9.1. If cp i s a bounded subfunction on IcJ, Cp has a t most a countable number of discontinuities on A t each to E I0 cP(tO) 5 max[T(ti),Cp(t,)l.

then I.

Proof:

The f i r s t assertion i s a c l a s s i c a l r e s u l t t h a t

follows from the f a c t t h a t

+

cp(to) 53

and

cp(ti)

e x i s t a t each

to

E

0

I

.

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

By Theorem 1.1.2 and t h e f a c t t h a t

i s a sub-

cp

function, t h e second a s s e r t i o n r e a d i l y follows. We consider next t h e d i f f e r e n t i a b i l i t y of subfunctions For a f'unction

with a f i n i t e r i g h t l i m i t

h(t)

h(t:)

.

at

to, we define

-

+

Dh(tO) = lim

+

h ( t ) h(t;) t-t 0

t+tO provided t h e l i m i t e x i s t s .

THEOREM 1 . 9 . 2 . then

to

E

Dcp(t:), 0

I

.

cp

Proof:

i s a bounded subfunction on

I c J,

e x i s t i n t h e extended r e a l s f o r each

I. A s i n Theorem l . g . l w e s h a l l consider only one

Suppose t h a t

lim i n f

+

t-tt

to

t

E

- to

Io and t h a t

< l i m sup t-tt

0

Let

Dh(ti).

The appropriate one-sided derivatives e x i s t a t f i n i t e

end points of

case.

If

Dcp(ti)

Analogously we define

+

-

cp(t> cp(to>

t

- to

0

m be a r e a l number s t r i c t l y between these two limits.

Then t h e i n i t i a l value problem x" = f ( t , x , x ' ) , has a solution

+

x(t,) = cp(tO), x ' ( t o ) = m

x E C ( ~ ) [ [ ~ ~ , ~ ~ f+o r~ some I , R I6

54

>

0.

AS

1.9. PROPERTIES OF SUBFUNCTIONS

there e x i s t

t 1, t2, t3 such t h a t

It follows t h a t

x(t2).

Since

cp

to < t < t2 < t < t 3

1

0

cp(t,) < x(t,), cp(t,) < x ( t 3 ) , and cp(t2) i s a subf’unction t h i s i s a contradiction

+

6,

>

and therefore we are done.

The c l a s s i c a l r e s u l t s i n the theory of r e a l variables give as a consequence the following corollary.

COROLLARY 1 . 9 . 2 . If cp i s a bounded subfunction on I c J, then cp has a f i n i t e derivative almost everywhere on I. LEMMA 1.9.1. If

cp

i s a subf’unction on

I

bounded above on each compact subinterval of

I,

i s a subfunction on

I.

y(t) = l i m

cp(s)

Proof:

Let

[tlyt21

C

J

and i s

then

I a n d l e t x ( t ) b e a s o l u t i o n o f (1.8.1)

with y(t,) 5 x(t,) and y ( t 2 ) 5 x ( t 2 ) . Then cp(ti) 5 y ( t i ) 5 x(ti), i = 1,2, and, because cp(t) i s a subf’unction,

5 x ( t ) on [t,,t,]. We therefore have f o r each tl < t < t2, y ( t ) = l i m sups+t cp(s) 5 x ( t ) . Hence y ( t ) 5

p(t) x(t)

on

[tl,t2]

and y ( t )

i s a s u b f b c t i o n on

I.

We now proceed t o discuss some l a t t i c e properties of subf’unctions.

THEOREM 1.9.3. Assume t h a t the s e t of subfunctions 55

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

on the i n t e r v a l I C J i s bounded above a t each I. Then cp,(t) = supCYEA cpCY ( t ) i s a subf’unction on

{cpCY: CY E A]

point of

I. I and x E C(2)[[tl,t2],R] Suppose t h a t [tl,t2] i s a solution w i t h cp,(t) < x ( t ) a t t = tl,t2. By the definition of q0, we then have cpCY(t)5 x ( t ) a t t = t t 1’ 2 f o r each CY E A . Since each cpCY i s a subfunction on I, it follows t h a t cpCY(t)5 x ( t ) on [tl,t2] f o r each CY E A . This implies cpo(t) < x ( t ) on [tl,t,] and cpo(t) i s a subfunction on I.

Proof:

IIHEOREM 1.9.4.

I&

cp be a subfunction on an i n t e r v a l

I1 w i t h I1= i1n I. Suppose further that cp,(t) 5 cp(t) a t f i n i t e end points of I1 which are contained i n I. Then cp2 defined by IC J

and cpl

a subfunction on

cp,(t> =

{

~“P1(t)>cp(t)l>

I1J

t

cp(t>,

E

I - 11,

i s a subf’unction on I. Proof: By h y p t h e s i s cp,(t) = cp(t) i s a s u b m c t i o n on I I1 and, by Theorem 1.9.3, cp2(t) i s a subfunction on I1. Consequently, we need only t o show t h a t we have the correct behavior on intervals [tl,t,] C I which are not contained i n e i t h e r I1 o r I-I1. Again l e t us consider j u s t one case since the arguments are similar. Assume tl E 11, t2 E I 11,

-

-

56

1.9. PROPERTIES OF SUBFUNCTIONS

i s a subf'unction on

In particular,

-

cp(t) < x ( t )

we obtain

I,

5

cp2(t) = cp(t)

x(t)

on

on

(t3,t2].

[t,,t,].

Also

'~l(t35 ) c ~ ( t 3 )_< x(t3) and cpl(t1) 5 ~ 2 ( t l 5 > x ( t l > , hence cp,(t) 5 x ( t ) on [ t t I because cpl(t) i s a subfunction 1'

3

cp2(t) = max[cp(t), cpl(t)] < x(t)

We thus i n f e r t h a t

on

I1.

on

[tl,t3].

Putting these things together we see t h a t

cp2(t) 5 x ( t ) on [tl,t2]. The other p o s s i b i l i t i e s a r e dealt with i n a similar way and we conclude t h a t cp2(t) i s a subfunction on

I.

MEOREM

(i) x

f

for each

1.9.5. E

Suppose t h a t

C[J xR x R , R ] , f ( t , x , x I )

i s nondecreasing i n

(t,xl)

and ( i i ) t h a t solutions of i n i t i a l value problems are unique.

Then, i f

$(t)

i s an upper solution on

cp(t> i s a bounded subfunction on and cp(t;> 5 $(t,>, we have

Proof:

Suppose t h a t

[tlJt2]

[tlJt2]

cp(t1.5 $ ( t )

cp(t) > $ ( t )

C

J

and

with cp(t1) on

5 $(tl>

(tl't2).

a t some points of

(tlJt2). Observe t h a t it i s enough t o consider the case where cp i s upper semicontinuous on (tl,t2). To see t h i s , l e t

cp(t) = l i m sup cp(s) on (t,,t,). Then by Lermna 1.9.1, s+t -cp(t) i s a subfunction on (t t ). By Corollary 1.9.1, 1' 2 - + + cp(tl>= cp(t,> 5 $(tl> and F(t; cp(t;> 5 J'(t2). mreover, rp(t) > @ ( t ) a t some points i n

(t t ).

at some points i n

rp(t;)

5

(tl,t2).

a t some points i n

$(t,)

implies

cp(t)

[to

- 6, to+ 61

6

>0

(tlJt2),

and 0

57

This being the case,

(tl,t2)

- $(t)

which i s assumed on a compact s e t Then there e x i s t s a

implies T ( t ) > q ( t )

As a r e s u l t we may assume cp(t)

1' 2 i s upper semicontinuous on

cp(t>> + ( t )

(tlJt2)

<

and

cp(tf)

5

$(tl>,

has a positive maximum M E C (tl,t2).

E

>0

&

< $(to+ E ) +

Let

to= lub E.

such t h a t M - 9(t0 + 6 ) ~

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and such t h a t the BVP

has a solution x

C(2) [[to

E

Since

by Theorem 1.1.2.

- 6,

$(t)

+

to+ 6],R]. This i s assured M i s an upper solution, we

+

x(t) < - $(t)

obtain by Theorem 1.2.3,

However ,

V(t0

- 6) 5 $(to - 6)

cp(t

+ 6) < +(to + 6) +

+

on

M

( t o - 6, to+ 6).

-

M = X(tO 6)

and

which yields

0

5 x(to),

cp(to)

subfunction on cp(to) = + ( t o )

+

M-

E =

X(tO + S ) ,

because of the f a c t

t0 We therefore conclude t h a t

(tlJt2). M.

This contradicts

E

cp(t) i s a E

-

q ( t ) < $(t)

(tlJt2).

On

and

EXERCISE 1.9.1. Prove t h a t the assertion of Theorem

1.9.5 remains valid

i f i n place of the assumption "that

solutions of i n i t i a l value problems are unique" we suppose that to

f (t,x,x')

XI

s a t i s f i e s a Lipschitz condition with respect

on each compact subset o f

J X R xR.

We s h a l l now discuss properties of bounded functions t h a t are subfunctions and superfunctions simultaneously.

We w i l l

need a well-known r e s u l t concerning solutions of i n i t i a l value problems which we merely s t a t e . LEMMA 1.9.2.

6

> 0, % >

0,

If

and

%>0

i n i t i a l value problem i s defined on

I

6

=

(tOJXOJxi)E J X R X R ,

such t h a t every solution of the

x" = f ( t , x , x ' ) ,

[to

- 6,

there exist

to+ 61 58

n

J.

x ( t o ) = xo, x f ( t o ) = xd Moreover,

1.9. PROPERTIES OF SUBFUNCTIONS

THEOREM 1.9.6.

Assume t h a t

f(t,x,xt)

i s such t h a t

C(2)

solutions of boundary value problems, when they exist, are unique. I

and t h a t

J

C

Suppose t h a t z(t)

i s bounded on each compact subset

z(t)

i s simultaneously a subfunction and a

superfunction on

I. Then z ( t ) i s a solution of x " = f ( t , x , x t ) on an open subset of I the complement of which has measure zero.

Furthermore, i f

z(t)

a t which

z(t)

to E Io i s a p i n t of continuity of does not have a f i n i t e derivative, then

either

+

Dz(tO) = Dz(t-) = + m 0

+

+

If z ( t o ) > z(t,)J D z ( t 0 )

+

DZ(tO) = Dz(t')

=

0

Proof:

-

+

or =

D z ( t ) = Dz(t-) = 0

Dz(t-) 0

=+m

0

and

z(ti)

-

W.

< z(t,),

W.

By Corollary 1.9.2,

z(t)

has a f i n i t e derivative

to E Io i s a point a t which z ( t ) has a f i n i t e derivative, there i s a 6 > 0 such t h a t almost everywhere on

[ t o -6, t 0 + 6 1 and

I.

I J Iz(t)l

If

5

Iz(to)l

1 On

[ t o - 6, t 0 + 6 1 ,

It then follows from Theorem 1.1.2 t h a t there i s a 0

<

45

6, X"

61

= f(t,x,x'),

has a solution x(t)

a solution of

0,

x

E

C(2)[[to

~ ( t , + 6 ~= ) ~ ( t ~ + 6 ~ ) ,

- 61,t0+61]JR].

Since

z(t)

a t the same time a subf'unction and a superfunction on z(t)

>

such t h a t the BVP

on

[to

- 61Jt0 + S1].

xtt = f ( t , x , x t ) 59

We conclude t h a t

on an open subset of

is

I, z(t) I

the

is

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

complement of which has measure zero.

to f 'I be a point of continuity of z ( t ) at which ~ ( t )does not have a f i n i t e derivative. By Theorem + 1.9.2 Dz(tO) and Dz(ti) both exist i n extended r e a l s . If Next l e t

both are f i n i t e , then by the same argument as used above there

i s an i n t e r v a l around to i n which This contradicts the assumption t h a t

Dz(t0)

Dz(t:)

=

+m

z(t)

i s infinite.

and Dz(ti)

# + m.

Then there e x i s t numbers

> w(t)

>0

f(t,X,X'),

x(tO

has a solution x1 there i s a

b2

solution on

>0

-

E

- 4)=

C(2)[[to

= f(t,X,X'),

41

W(tO

such that the x(to) =

Z(to),

1.9.2, can be extended t o be a

x,(t)

By Lemma

and such t h a t a l l solutions of

x ( t o ) = Xl(tO)'

[ t O , t O+ 6*1

- 61)y

- 61,tO],R].

such t h a t

[to 61 ,t 0 + the i n i t i a l value problem

e x i s t on

-

4, 0 < 61 5 6,

BVP

XI1

6

on to 6 5 t < to.

= z(to) + N ( t -to)

By Theorem 1.1.2, there i s a

=

does not have a

such t h a t

and N

XI1

z(t)

to. Consequently, a t l e a s t one of To be specific, suppose

f i n i t e derivative a t Dz(t:),

i s a solution.

z(t)

and s a t i s f y

x'(to) =

lx(t)

X 1' ( t

0 ),

- x(to) I 5 M ( t - t o )

[ t 0 , t 0 + 6 2 ] y where M = I x ' ( t ) I + 1. Again applying 1 0 Theorem 1.1.2 we i n f e r t h a t there i s an q, 0 < 7 5 min[fj ,6 1, 1 2 such t h a t f o r 0 < 6 < q the BVP

on

3-

X" = f(t,X,X'),

X(tO

- 63 ) = Xl(tO - 63),

x ( t O+ 6 ) = Xl(tO)

3

60

+

(M+ &)6

3

1.9. PROPERTIES OF SUBFUNCTIONS

has a solution E

> 0.

that

As 0

<

x2

- 63 , t0 + 63 ],R]

f o r some fixed

E C ( 2 ) [[to

4

Dz(ti) = + m, we can assume t h a t i s chosen s o 5 q and xl(tO) + ( M + & ) 6 < z ( t 0 + 6 ). Then

4

3

3

xl(tO + 6 ) < x 2 ( t 0 + 6 )

3

3

< z ( t O +6 ) 3

and

- 9'

Xl(tO

=

X2(t0 - 6+

5

- 4,.

z(tO

The l a s t inequality follows from the f a c t t h a t

- 9)

~ ( t )i s a

- 4).

superfunction and xl(tO) = z ( t o ) , xl(tO < z(tO Since ~ ( t )i s a superfunction, we deduce from the above inequalities t h a t

-< z ( t )

x2(t)

on

[to

- 63 ,t0 + 63 ].

From the

same inequalities and the f a c t t h a t the solutions of BVPIs, when they exist, are unique, we also i n f e r t h a t on

[to - 6 ,t + 6 3

3I.

0

x i ( t o ) = x;(to),

Thus

hence,

xl(t) 5 x 2 ( t ) x (t ) = z ( t o ) = x2(t0) and 1 0 x 2 ( t ) i s a solution of t h e i n i t i a l

value problem with the i n i t i a l conditions x l ( t o ) = x ' ( t ). 1 0

However,

Ix2(to +

4,- X2(t0) I

=

x(t,) = xl(tO),

(M+ €183

which contradicts the f a c t t h a t a l l solutions of t h i s i n i t i a l Ix(t) - xl(tO) I 5 M ( t - t o ) on We are forced t o conclude t h a t 'Dz(t;) = + m.

value problem s a t i s f y

[ t O , t O+ 4 1 . By similar arguments using t h e f a c t t h a t

z(t)

i s also a

subfunction, the other statements regarding the behavior of a t a point of continuity can be established.

z(t)

We consider now the behavior of If

z ( t ) i s bounded on each compact inter+ Furthermore, z ( t o ) # z ( t 0 ) since, by Corollary

and are f i n i t e since Val of

a t p i n t s of

to E Io i s a p i n t of discontinuity of and z ( t i ) both e x i s t then by Theorem 1.9.1, z(t:)

discontinuity. z(t),

z(t)

I.

1.9.1, 61

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

+ > .(ti)

Assume t h a t

is a z(t)

6

z(to)

>0

< w(t)

and an N =

= f(t,x,xf),

has a solution Xl(t0) = z(t:) and

z(t)

x

2

Dz(ti)

6

Then t h e r e

[tO,tO + 61 C I and on to < t 5 to + 6. By

-to)

4,

0

<

45

such t h a t t h e BVP

6,

x(tO+fjl) = w ( t O + 61),.

x(to) = z(ti), 1

# + m.

such t h a t

+ z(to) + N ( t

Theorem 1.1.2 t h e r e i s a XI'

and t h a t

C ( 2 ) [ [ t o , t o + 611,R].

x1 (t0 '61)

z(to),

i s a subfunction,

x,(t)

= w(to+61)

1 z(t)

on

>

z(to+61),

[to,to+61].

Now proceeding as i n t h e previous paragraph and using the f a c t that

z(t)

i s a subfunction, we can obtain a solution of t h e

i n i t i a l value problem XIt = f ( t , X , X ' ) ,

x ( t o ) = Xl(tO),

X'(t0) = x;(to),

t h e graph of which i s not contained i n t h e sector t o t h e l e f t of

to in which such solutions must be.

diction, we conclude t h a t

+ Dz(tO) = +=J.

From t h i s contraThe other assertions

concerning derivatives a t points of discontinuity of are d e a l t with i n a similar w a y .

z(t) The proof of t h e theorem i s

now complete. 1.10

PERRON'S METHOD mploying t h e properties of sub-

and superfunctions and

the existence " i n the small" theorem, we consider the existence

i n the large f o r t h e boundary value problems by Perron's method. DEFIXITION 1.10.1.

A bounded real-valued function

62

1 .lo. PERRON'S METHOD

defined on J

i s said t o be an underfunction with respect t o

the BVP (1.10.1)

x" = f ( t , x , x ' ) ,

where

C[J X R xR,R],

f E

i s a subfunction on J

.(a)

= A,

i n case

v(a)

x(b) = B,

5 A,

cp(b)

5

and cp

B,

r e l a t i v e t o (1.8.1). An overfunction

i s defined similarly i n an obvious way. THEOREM 1.10.1.

x"

=

f(t,x,x')

unique.

Assume t h a t solutions of BVP's f o r

on subintervals of

when they e x i s t , are

J,

Suppose t h a t there e x i s t both an underfunction cpo

and an overfunction $ r e l a t i v e t o BVP (1.10.1) such t h a t 0 cpo(t) 1. q0(t) on J . Let Q be the s e t of a l l underfunctions cp

such t h a t

cp(t) < $o(t)

i s simultaneously a sub-

function on J.

a solution x

5

E

z(t,),

W e define

Then

z ( t ) = sup

and superfunction on J.

It follows from Theorem 1.9.3 t h a t

Proof:

subfhnction on J .

x(t2)

on J .

zl(t)

Suppose now t h a t

z(t)

but

x(t)

>

such t h a t

z(t)

[tl,t2]

is a C J

x(tl) < z(tl),

a t some points of

LtlJt21

on J Then by Theorem 1.9.4, zl(a) = z(a)

zl(t)

-< A,

- Ctl,t21.

i s a subfunction on z (b) = z(b) 1

-< B.

and

(tl,t2).

on J by

Zl(t) = { ~ ~ ~ ( t ) J z ( t ) l on

kreover

z(t)

i s hot a super-

Then there i s a subinterval C(2)[[tlyt2],R]

cp(t) v=Q

J

and

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and

jro

[tl,t2].

a superfunction implies t h a t Consequently, we have

we i n f e r t h a t x(t)

> z(t)

which proves

x(t)

5

z,(t)

5

jro(t)

jro(t)

on

on Hence,

J.

E 0 and z,(t) < z ( t ) . However, z1( t ) = 1 a t some points i n (t,,t2), a contradiction

z

i s a superfunction on

z(t)

J.

This proves

t h e theorem. The function

DEFINITION 1.10.2.

obtained i n t h e

z(t)

preceding theorem depends on t h e BVP (1.10.1) function

We s h a l l say t h a t

jr0.

z(t)

and on t h e over-

i s a generalized

solution of the BVP (1.10.1) and we s h a l l designate it by z(t;JIO). Notice t h a t since

z(t;jr ) 0

i s a subfunction and a super-

function a t t h e same time, the assertions made i n Theorem

1.9.6 apply t o z(t;Jro). We therefore have t o consider the behavior of

a t t h e end points of

z(t;jro)

J = [a,b].

We

discuss t h i s i n t h e following theorem. THEOREM 1.10.2.

Assume t h a t the hypotheses of Theorem

1.10.1 are s a t i s f i e d and l e t

z ( t ) = z(t;Jro) be the corresponding generalized solution of (1.10.1). Then .(a) = A. If

Dz(a+)

Hence, i f

z(a+)

#+m,

Dz(a+)

-< .(a).

i s finite,

t

assertions are t r u e a t Proof: cp*(a) = cp(a)

cp*(t)

+

c, c

cp(t)

Assume now t h a t z(t)

6,0

Dz(a+)

< 6 0,

< A, Dz(a+)

z(a+) = z ( a ) = A.

Observe t h a t i f

the function

z(a+)

If

z(t),

# +m

and

and an

+ N(t 64

yields

-a),

N

z(a+)

(a,bl

This, t o -

z ( a ) = A.

>

~ ( a ) . Then

such t h a t a

and

< t 5 a + 6.

1.10. PERRON'S METHOD

By Theorem 1.1.2 it follows t h a t f o r

0

<

E

< z(a+) -.(a)

and 0 < 6 < 6 s u f f i c i e n t l y small, t h e BVP 1-

has a solution x

E

C(2)[[a,a+61],R]. z ( t ) _< x ( t )

f'unction, we have z(a+)

5

on

x(a+) = .(a)

This contradiction proves t h a t Finally, suppose t h a t

[&,a+ = z(a+)

Dz (a+)

Z(a>* z(a+)

Since

41

z(t)

i s a sub-

which yields

- E.

# +m

implies

< A and Dz(a+)

# - m.

using an analogous argument as above and the f a c t t h a t

i s also a superfunction on J, implies

-

Dz(a+) =

If Dz(a+)

z (a+) 5

we can conclude t h a t

Then

~ ( t ) z(a+) < A

m.

i s f i n i t e , then combining the assertions of

t h i s theorem we arrive a t

A = .(a)

i s complete.

> A. -> z(a+) -

The proof

From the preceding r e s u l t s it i s clear t h a t Perron's method of studying t h e BVP (1.10.1) canbe separated i n t o twoparts. The f i r s t p a r t dealswiththeproblemofestablishingtheexistenceof an overfunction qo and an underfunction cP,(t) 5 q o ( t )

on J.

rp 0

such t h a t

The second part consists of finding

conditions under which the generalized solution of class

C(2)[J,R]

and i s a solution on J .

z (t;qo)

is

I n view of

Theorems 1.9.6, 1.10.1, and 1.10.2, accomplishing t h i s i s equivalent t o showing t h a t Dz(t-)

i s f i n i t e on

Dz(t+)

i s f i n i t e on

[a,b)

and

(a,b].

Let us f i r s t give s u f f i c i e n t conditions f o r the existence

of under-

and overf'unctions

LEMMA 1.10.1.

nondecreasing i n x

Let

.

f E C[JxRxR,R]

f o r fixed 65

(t,x').

and f ( t , x , x ' ) be Assume t h a t f ( t , x , x ' )

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

i s such t h a t lower and upper solutions of the d i f f e r e n t i a l equation x'' = f ( t , x , x 7 ) are subrespectively.

and superfunctions,

Then,

i s a solution of XI' = f ( t , x , x I ) (a) i f u E C ( 2 ) [ J , R ] on J, there e x i s t overf'unctions and underf'unctions r e l a t i v e t o any BVP on J; (b)

i f there i s a

(1.10.2)

k

>0

such t h a t

5

If(t,O,x') -f(t,O,O)l

t

klx'),

E J,

XI

E

R,

there e x i s t overfunctions and underfunctions r e l a t i v e t o any BVP on

J.

For any s u f f i c i e n t l y large M > 0, q o ( t ) = u ( t ) + M i s an overfunction, rpo ( t ) = u ( t ) M i s an underfunction, and

Proof:

cpo(t)

5

q 0 ( t ) on J.

-

This proves ( a ) .

To prove (b), we l e t

sider the solution w(t)

M = maxlf(t,O,O)l

5

M

>0

q 0 ( t ) = w(t)

v" = - k v ' v(t)

5

0

As a

on J

+

and we a s s e r t t h a t ,

M w i n be an overfunction

v ( t ) be the solution of BVP

+ My

and v ' ( t )

that dt = klv'

-< f(t,w,w')

i s s u f f i c i e n t l y large.

Similarly, l e t

on J .

wf(b) = 0.

and w t ( t )

f(t,O,wt)

f o r a given BVP, provided

Then

w(a) = 0,

-> 0 on J. Hence, by (1.10.2), = - kw' - M = - klw' I - M < f (t,O,w') - f (t,O,O) - M. 410

Then w(t)

Thus w"

and con-

of BY!?

w" = - k w * - M y

W"

on J

v(a) = 0,

-< 0

on J,

v'(b) = 0. from which it follows

I +M> f(t,O,v') -f(t,O,O) + M > f(t,O,v') r e s u l t , vl' > - f ( t , v , v * ) on J and, f o r M > 0

s u f f i c i e n t l y large,

cp,(t) = v ( t ) 66

-M

i s an underfunction

1.10. PERRON'S METHOD

relative t o a given BVP. Obviously, we have q 0 ( t ) on J. !Chis completes the proof of the lemma.

5

qo(t)

We are now ready t o s t a t e our main r e s u l t . THEOREM 1.10.3.

Assume t h a t

i s nondecreasing i n x

f o r each

f

Q

(t,x'),

of i n i t i a l value problems are unique. that there i s a

k

C[JxRxR,R], f ( t , x , x ' )

> 0 Such t h a t

and t h a t solutions

In addition, suppose

I f ( t J 0 J x ' ) -f(tJOjO)l < klx'

t E J and x t e R . Then f o r any BVP on J with an associated overflrnction q0 ( t ) , the generalized solution 0 Z(t) = Z(t;qo) E C(2)[Jo,R] and Z" = f ( t J ? d J Z ' ) On J for

.

Proof:

By Lemma l . l O J I J we readily obtain the existence

rpo

Jr0 with respect - JIo(t) on J . Consequently, t o BVP (1.10.1) such t h a t cpo(t) < the generalized solution z ( t ) = z(t;q0) i s defined. b r e over, the hypotheses imply t h a t solutions of BVP's when they exist, a r e unique and, therefore, the conclusions of Theorem 1.9.6 apply t o z ( t ) . Thus it suffices t o show t h a t Dz(t+), of an underfunction

and an overfunction

0

0

.

Let to E Jo and Dz(t0) a r e f i n i t e f o r every to E J + suppose t h a t z ( t O ) 2 z ( t i ) . The alternative case can be dealt with i n a similar way and w i l l not be t r e a t e d . We break the discussion i n t o two p a r t s .

F i r s t assume

that z ( t i ) 2 0. Let + ( t ) be a solution of Jr" = - kJr' -M, where M = max If(t,O,O)l, with Jr(to) = 0 , + ' ( t ) _> 0 on [tOJb], and

J

Jr(b) _>z(b-), which can be computed.

in Lemma 1.10.1, +,(t)

=

q(t)

+ + z(to)

on [t0'b1 w i t h z(tA) _< ql(tO) and follows from Theorem 1.9.5 t h a t z ( t )

$ ( t ) be a solution of

i s an upper solution

z(b-) Jr,(t)

_< Jrl(b). on

It (tOJb),

q ' ( t ) < m. Applying Theorem 1 0 z ( t ) i s continuous a t to. Now l e t

which implies t h a t Dz(tt)

1.9.6, we a s s e r t t h a t

Then as

Jr"

5

= kJr'

67

-M

on

[a,to]

such t h a t

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

$ ( t o ) = 0 , $(a)

2

z(a+),

$'(t) <0

and

again by Theorem 1.9.5, we have Dz(t0) 2 $ ' ( t o ) > - m. z ( t ) has a f i n i t e derivative a t implies

Finally, suppose t h a t solution of

cpl'

z(a+), cp(to) = 0, Theorem 1.9.5,

kcp'

=

+

and

on

2

<

0

0

on

cp,(t) be a

and l e t such t h a t [a,tol.

cp(a)

Theorem 1.9.6 t h a t we can show t h a t

z(t)

i s continuous a t

+ > - m.

5

Then, by

,> cp,(t) - z ( t 0 ) on (a,to), Dz(t0) 5 cp;(to) < m. It follows

z(t)

assures us t h a t

Then

to.

[a,to]

cp'(t)

[a,tol.

z ( t ) 5 $ ( t ) + z ( t o ) , which We i n f e r t h a t i n t h i s case

.(ti)

M

on

which from

to. Similarly

Thus we again conclude t h a t

Dz ( t o )

to. z ( t ) has a f i n i t e derivative a t each point of

z ( t ) has a f i n i t e derivative a t J

0

,

Since

it follows from !Theorem 1.9.6 t h a t z E C(2)[Jo,R] and 0 z" = f ( t , z , z ' ) on J This completes the

.

i s a solution of proof.

The following exercise i s instructive. EXERCISE 1.10.1.

Under the hypotheses of Theorem 1.10.3,

show t h a t the BVP x" = f ( t , x , x ' ) , x(a) = x(b) = 0 has a unique solution x ( t )

where M,k

on J,

Hints:

- k$, - M)

If

such t h a t

as i n Theorem 1.10.3.

$,(t),$2(t)

are the solutions of BIT'S

9; = k$, -M, q2(b) = 0, = 0, show t h a t Jrg(t) = min[$,(t),$,(t)] i s an overfunction. Similarly obtain an underfunction cpo(t) 5 $ o ( t ) $'I

=

1 $;(a)

on J . cpo(t)

5

Jrl(a) = 0, $;(b)

= 0,

Using Theorems 1.10.2 and 1.10.3, proceed t o show that z(t;q0)

5

q 0 ( t ) on J 68

and

z(t;q0)

i s a solution of

1.1 1 . MODIFIED VECTOR FUNCTION

t h e desired BVP which i s unique.

Compute

qoJq0 and est a b l i s h t h e bound on I x ( t ) l ; show t h a t m e x Jr ( t ) = J O -minJ cpo(t>. If to E JO and x ( t o > > 0 , l e t q1(t;t0)J q 2 ( t' - 0t ) be the respective solutions of BVP's

- kJr' -M,

$1

=

+"

= kJr'

- M,

q(t,) = 0,

$'(b) = 0;

+ ( t o ) = 0, $ ' ( a ) = 0 .

prove t h a t

q$(to;tO) ,< x ' ( t O ) 5 Jri(t,;t,),

x ( t o ) < 0,

obtain analogous relations.

Similarly, i f

Since the functions

involved and t h e i r derivatives can be computed, deduce the desired bound on

Ix' ( t ) I.

1.11 KlDIFIED VECTOR FUNCTION

Some of the r e s u l t s considered so f a r f o r scalar secondorder d i f f e r e n t i a l equations w i l l now be extended, i n the sections t h a t follow, t o f i n i t e systems of second-order difTo avoid repetition, l e t us agree on

f e r e n t i a l equations. the following:

..,n;

1,2,.

the subscripts

range over the integers

i,j

vectorial inequalities w i l l be used freely with

the understanding t h a t the same inequalities hold between t h e i r respective components. L e t us consider a system of second-order d i f f e r e n t i a l

equations written i n the vector form (1.11.1)

where

XI1 =

n

f E C [J xRn X R

,Rn 1.

f(t,X,X'), We s h a l l next define modified

vector functions i n different ways. DEFINITION 1.11.1. Let a(t) that

5

p(t)

on J

IcY'(t)l,

cr,B

E

C(l)[J,Rn] with

and l e t a constant vector

\pl(t)( < c F*(t,X,X')

on J . =

69

Then define

f(tJx,z')

c

>0

be such

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

("s

where

-x!J

=

if

J if -c < x! j - J if x!

-C

J

Fr((t,f;,x')

,x,

,x,x') =

XI

+ (xi

-

>c 3' 5 cj , < -c 3'

2 B i ( t ) ) / l + x i if xi

>

Bi(t),

Fy ( t x ) if a i ( t ) 5 ~ i ( ~ i ( t ) , 2 F y ( ( t , x , x ' ) + (xi - a i ( t ) ) / l + x i if xi < a i ( t ) ,

where

The f'unction f(t,x,x')

will be c a l l e d t h e modification of

F(t,x,x')

relative t o the t r i p l e

the definition t h a t J xRnxRn.

a,@,c.

It i s c l e a r from

i s continuous and bounded on

F(t,x,x')

IF'I ,< c and a ( t ) ,< 5 5 @ ( t )

Also note t h a t

on J . DEFINITION 1.11.2. on

J

A,

where, as before,

and l e t

cp,$

E

Let

a,@

E

C[J,Rn]

with

F ( t , x , x ' ) = f(t,x,f;'),

-(.

Bj(t)

x3

j

aj(t)

< B(t)

such t h a t cp(t,x) 5 q ( t , x ) on A = [(t,x): a ( t ) <x< @ ( t ) , t E J].

C[A,Rn]

men define

where

a(*)

if x.J

>

Bj(t), if a . ( t ) < x . J;i(t,x),

Jrg(t,X)

if xj'

x'

if cp.(t,x)

{ scpj(t,x)

5 x!J 5 Jlj(t,X), J if x! < cpj(t,x). J

We shall c a l l the f'unction F(t,x,x'), the modified f'unction o f f ( t , x , x ' ) associated w i t h a,@,cp, and $. fgain, it i s easily seen t h a t F(t,x,x') i s continuous and bounded on JxRnxRn.

Furthermore,

a(t) <

x -< @ ( t ) and

cp(t,x)

5

5

$(t,x>* DEFINITION 1 . 1 1 . 3 .

Let

E = [(t,x,x'): t

+ +

E

J,

C[R ,R 1 be such that 6(s) = 0 accordingly as 0 5 s respectively. Then define Let

6

E

N > 0 be given and l e t

p,

(Ix((

C P, X'

E

Rnl.

c

6(s)

< 1, and 5 N, N < s < 2N, and s > 2N 6(s) = 1, 0

We s h a l l sey that F ( t , x , x ' ) i s a modified function of f ( t , x , x ' ) r e l a t i v e t o p,6. Clearly the f'unction F ( t , x , x ' ) n n i s continuous and bounded on J xR xR

.

A r e s u l t analogous t o Theorem 1.3.1 w i l l be proved next.

where

71

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

and I a ' ( t ) l J I p ' ( t ) l < c. Suppose f u r t h e r t h a t n n f E [ J X R ~ X ,R R I and f ( t J x , y ) i s quasimonotone nonincreasing i n

t h a t is, f o r fixed

x,y,

increasing i n y

for j is nonincreasing i n x t h e modification of

j

f

and f o r fixed

i

for

(t,x), f i ( t , x , y )

j

#

f(t,x,x')

t h e BW

(1.11.2)

x" = F ( t , x , x ' ) , x

has a solution

E

5y5

.(a)

x ( a ) = y,

C(*)[J,Rn]

is

F(t,x,x')

associated with

cording t o Definition 1.11.1, and 6 5 p(b),

(t,y), fi(t,x,y)

Then, i f

i.

i s non-

a,p,c

ac-

p(a), a ( b ) 5

x ( b ) = 6,

satisfying

a(t)

-< x ( t )

_< p ( t )

on J . Proof:

By Corollary 1.1.1, t h e BVP (1.11.2) has a solution.

Hence we need only t o show t h a t

w i l l show t h a t x(t)

5

p(t)

a(t)

on

5

x(t)

5

a(t)

on J,

x(t)

5

p(t)

on J .

since t h e proof of

follows s i m i l a r l y .

J

Suppose it i s not t r u e t h a t t h e r e e x i s t s an index k

a(t)

_< x ( t ) on J.

and an i n t e r v a l

Then

[ t l J t 2 ] C (a,b)

such t h a t 5 ( t 2 > = ak(t2)J

%(tl> = ak(tl), and 5(t) Thus

ak(t) -\(t)

%(to) = a i ( t o ) .

< ak(t>J

(tlJt2)'

has a maximum at some Hence

I

/%(to)< ck

have

72

to

E

(tl,t2)

and

and consequently, we

We

1.1 1 . MODIFIED VECTOR FUNCTION

i n view of t h e d e f i n i t i o n of nonincreasing character of possible at a maximum of

F(t,x,x') f(t,x,x')

cik(t)

and t h e quasimonotone

x,x'.

in

- %(t)

and hence t h e proof.

The quasimonotone nonincreasing nature of x,x'

This i s im-

in

f (t,x,x')

assumed i n Theorem 1 . l l . l b e c o m e s superfluous i f t h e

assumptions concerning

a,p

are made stronger.

"his we s t a t e

i n the following exericse. EXERCISE 1.U.1. Suppose t h a t with

a(t)

5

p(t)

on

J

and f o r

fi( t,A(t,i),At ( t , i ) ) , D-p;(t) A(t,i)

,

=

A ( t i) =

5

n 2

a,p E CIJ,Rnl

t

E

J

0

, D-a;(t)

C(l)[Jo,Rn]

fi( t , B ( t , i ) , B ' ( t , i ) ) ,

( X1' .. ., x i - l > q t ) , x i + l , ( xi, .. .,x i - 1,ai ( t ) ,xi+1,

*

where

.,xn),

...,xn ),

...7xi-19Pi(t),xi+l,. . .,xn), B 1( t , i ) = ( x i , .. . p i (t),xi+l, .. n ) ,

B(t,i)

= (xl,

.,XI

provided a . ( t ) < x . < p . ( t ) and - c . < x! 5 c f o r j # i, J - J - J J J j c being any vector s a t i s f y i n g I a ' ( t ) l , l p t ( t ) l < c . Show .(a)

-< y < p(a),

p(b) has a solution x E C(2)[J,Rnl

satisfying

that t h e BVP (1.11.2) such t h a t

73

a(b)

-<

6 _<

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

a(t)

-< x ( t ) 5

B(t)

on J.

REMARK 1.ll.1.

From t h e foregoing discussion, a n a t u r a l

question t h a t a r i s e s i s whether a r e s u l t analogous t o Theorem 1.11.1remains t r u e i f

a,B

ww,

solutions, i n a n a t u r a l

equation x" = f ( t , x , x ' ) .

a r e defined as lower and upper r e l a t i v e t o the vector d i f f e r e n t i a l

This can be done by defining a

modified function which has a jump discontinuity along t h e upper and lower solutions.

EXERCISE 1.U.2.

Let

x E C(2)[J,Rn]

be a solution of

xff = F ( t , x , x ' ) ,

where F i s the modified function of n n f E C[JxRnxR ,R ] r e l a t i v e t o p,6 as i n Definition 1.11.3,

Ilx(a) 1 , Ilx(b) 11 5 p. Suppose t h a t there e x i s t s a V E C[D,R+] such t h a t V(t,x,y) i s l o c a l l y Lipschitzian i n such t h a t

( x , ~ ) ,V(t,x,y) = 0 if llxll = P, V(t,x,y) > 0 if IIxll > P + and D VF(t,x,x') 2 0 in t h e i n t e r i o r of D, D being t h e set on

D = [[a,bl

x [x: llxll > P I xRn].

Show t h a t

Ilx(t)II

-< P

[a,b].

1.12 NAGUMO1s CONDITION

(CONTINUED)

I n remark 1.4.1, we noted t h a t Nagumo's condition i s not suff i c i e n t t o conclude

IIx' ( t ) 11 i s bounded, i f

n

f E C[J xRn xR ,R

n

Nonetheless, we can make conclusions on t h e a r c length of t h e curve

Ilx(t)II,

that i s

/ / / x ' ( s ) l / ds,

LEMMA 1.12.1. Assume t h a t

h E C[R+, (0,m) (1.12.1) where

.few

Ilf

1.

as the next lemma shows.

n n f E C[JxRnXR ,R ]

and

Suppose t h a t

(t,x,x')

11 5 h ( IIx' [I),

s ds/h(s) =

m.

(t,x,x')

E

J xRn xRn,

Then, t h e r e e x i s t s a 74

y: R' - t R +

1.

1.12. NAGUMO'S CONDITION (CONTINUED)

Proof: Since lorn s ds/h(s) y: R + Z satisfying

=

there e x i s t s a f'unction

rn,

From assumption (l.I2.1), we obtain

Hence i f

-

a < v, w < b,

by the change o f variables are equal (and zero) f o r respect t o w

((XI

v=w,

(s)II

--f

s;

the two integrals

and the derivatives with

e x i s t almost everywhere and are equal.

follows t h a t the integrals are equal since

Ilx'(w)II

It

is

absolutely continuous. Choose v and w such t h a t 11x' (v) 1) = el/ (b-a) (which i s possible by the mean value theorem) llxt(w)(I = max Ilx'(s)II on a 5 s 5 b . The desired r e s u l t then follows from (1.l2.2) and ( 1 . 2 . 3 ) and t h e proof i s and

complete.

It i s , however, possible t o extend Theorem 1.4.1 t o deal with the vector s i t u a t i o n . DLEORFM 1.12.1.

Suppose t h a t the following assumptions

hold: 75

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

n n ( i ) f E C[JxRnxR ,R 1, a , p E C[J,Rn] such t h a t a ( t > 5 p ( t ) on J and f o r t c J, u ( t > 5 x 5 p ( t ) , x f E R",

(1.l2.4)

5 hi(lX11),

Ifi(t,X,X')l

+

h, E C[R ,(O,m)]

(ii)

...,n,

i = 1,

and s a t i s f i e s

) - min a, ( t ) , where

Then, f o r any solution a(t) N

2

5

5

x(t)

@ ( t ) on

(1.u .6)

J

Ixf(t)l

N

E

C(2)[J,Rn]

of (1.11.1) with

t h e r e e x i s t s a constant vector u,p,h,

-< N

such t h a t on J .

I n view of ( l e a . ? ) , t h e r e e x i s t s a constant

Proof: vector

x

depending only on

0,

I.

- a ) = max[Ia(a) - @ ( a )

h(b

>

0

such t h a t

If (1.12.6) i s not t r u e , we m q r assume, without loss of

generality, t h a t t h e r e i s an index

t

J, I s ( t ) have E

I > Nk.

to = t O ( k )

\

E

such t h q t f o r some

However, by t h e mean value theorem, we

- s ( a >= where

k

(a,b)

x$t0)(b

- a),

and consequently, it follows t h a t

I%(tO) I 5 < Nk. The remainder of t h e proof i s almost identical t o the proof of Theorem 1 . 4 . 1 and we therefore leave it t o the reader.

76

1.12. NAGUMO'S CONDITION (CONTINUED)

The next r e s u l t i s an extension of Theorem 1.4.3 t o the present situation. TE30REM 1.12.2. (i) (ii) (t,x)

E

C(l)[J,Rn]

with

a(t)

'p,+

E

C(')[A,Rn]

with

cp(t,x)

+ 'px(t,x)

'pt

+t (iii)

- 5 f(t,x,'p(t,x)),

*

Jrx(t,x> * + ( t , x )

f E C[JxRnxRn ,Rn 3

2

and

f(t,x,+(t,x)); cp(a,x(a))

< x ' ( a ) _<

-

+(a,x(a)) * Then, f o r any solution x a(t)

5 x(t) 5

(1.12.7)

p(t)

on J,

cp(t,x(t))

Proof:

Let

C(2)[J,Rn]

E

we have

5 x'(t>5

x(t)

of (l.ll.1) such t h a t

Jr(t,x(t)),

t

E J.

be any solution of (l.U.1) satisfying

'p(a,x(a)) _< x ' (a) 5 +(a,x(a)). We s h a l l only prove t h a t p ( t , x ( t ) )

-< x ' ( t )

on J,

since the proof of the other case i n ( l . X . 7 ) i s similar. Assume t h a t t h i s i s not t r u e . and a

on

to

E

(a,b]

such t h a t

Then there e x i s t s an index k X;c(to) < cpk(to,x(to)). Set

(tl,to]. Such an i n t e r v a l

'pk(t,x(t)) -X;c(t)

>

0.

(tl,to] e x i s t s where

It then follows t h a t

77

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

on

(tl,to].

5

zi(t)

t

0

This together with assumption ( i i ) yields on

(tl,to]

increases.

Thus

and hence zk(tl)

tended t o conclude t h a t

>

0.

z,(t)

This argument can be ex-

>0

z,(a)

i s nonincreasing as

which implies t h a t

cp,(a,x(a)) > %(a), a contradiction t o assumption ( i i i ) . This proves cp(t,x(t)) 5 x l ( t ) on J and the proof i s complete. Another r e s u l t t h a t may be u s e h l i n some s i t u a t i o n s i s t h e following. THEOREM 1.12.3.

(i) .(a)

a,p E C(l)[J,Rn]

(ii)

#

with

a(t) < p(t)

h E C[JxRy,R~], h ( t , u )

i

where

t h a t is,

u,

t

f o r each

( 1.12.8)

E J,

and

u ( a ) = A,

A = max[la'(a)), l p l ( a ) l ] ,

for

t

E

J, a ( t )

5x5

I

e x i s t s on

> 0,

x(t)

5

B(t)

on J,

[a,b];

p(t),

5 h(t,

Then f o r any solution x E C(*)[J,Rn]

5

i s nondecreasing i n u

and t h e maximal solution of

If(t,X,X')

N

J

i s quasimonotone non-

hi(t,u)

u 1 = h(t,u),

(iii)

a(t)

on

= B(a);

decreasing i n j

Suppose t h a t

Ix'

I). of ( l . l l . 1 ) with

t h e r e e x i s t s a constant vector

depending only on a,@,h such t h a t I x l ( t ) l IN,

t

E J.

j'

1.12. NAGUMOS CONDlTlON (CONTINUED)

Proof:

a(t)

5

that

Let

5

x(t)

.(a)

x(t)

be any solution of (1.11.1)with

@ ( t ) on

= p(a),

This, i n view of t h e assumption

J.

implies t h a t

fine m(t) = l x ' ( t ) l .

5

al(a)

Then m(a)

-< A

D+m(t) < Ix"(t)I = I f ( t , x ( t ) , x ' ( t ) ) l

xl(a)

5

p'(a).

De-

and, by ( i i i ) ,

5

t

h(t,m(t)),

E

J.

Consequently, by t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s , we have m(t) where

[a,b].

5 N,

r(t,a,h)

t

r(t,a,A),

J,

E

i s t h e maximal solution of (1.12.8) which

r(t,a,A)

exists on

5

t

Let

E J.

N

>0

be a vector such t h a t

Then t h e s t a t e d conclusion follows

immediately. One can a l s o deduce a bound on Wapunov-like method.

IIx'(t))I by employing a This i s t h e content of t h e next two

theorems. THEOREM 1.12.4. (i)

in x,y, bl(u)

V

CID1,R+],

E

as

V(t,x,y)

D1 = J X[x:

where

+ w

Assume t h a t

u

+ w

( ( ~ 1 1-<

i s l o c a l l y Lipschitzian

5 b2( IIYII 1,

+

( i i ) D V(t,x,x') = lim sup h+O

(t,X,Y) +

+

D1;

(t,x,x')

and t h e m a x i m a l solution

of exists on

E

- V(t,x,x')]

-< g ( t , V ( t , x , x ' ) ) f o r

C [ J x R ,R]

u'

+ + 1,

C[R ,R

( l / h ) [V(t + h , x + h x ' ,

x' + h f ( t , x , x ' ) )

E

E

and

b l ( I h l l ) ~V(t,X,Y)

(iii) g

bl,b2

p] XRn,

u ( a ) = A ? 0,

= g(t,u),

[a,b].

79

E Int D ~ ;

r(t,a,h)

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Then, f o r any solution

x E C(2)[J,Rn] of (1.11.1)such t h a t

llx'(a)II

5

pending only on

p, po,g

such t h a t

Proof:

x(t)

Ilx(t)((_ 0 de-

be any solution of (1.11.1)such t h a t

5

llx'(a)II

Define

p0.

m(t) = V ( t , x ( t ) , x ' ( t ) ) .

Then by ( i ) , we obtain m(a) 5 b2( IIx'(a)lI) 5 b2(po). h = b2(po). Also, because o f t h e assumptions on V,

Choose

it

follows t h a t ~+m(t5 > @;(t,m(t>),

t

E

(a,bl

and consequently, by t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s , we obtain m(t)

-< r ( t , a , h ) ,

t

E J.

By ( i ) , t h i s i n t u r n yields

< r(t,a,h), bl( I I x l ( t > I I ) 5 V ( t J x ( t ) , x ' ( t ) ) = m(t> Since that

r(t,a,h) r(t,a,A)

e x i s t s on

5

M on

[a,b],

[a,b].

there is a

t

M

Furthermore, as

E

J.

> 0 such

bl(u)

--3m

> 0 such t h a t M < b(N). These considerations imply t h a t IIx' ( t ) 11 5 N on J . Clearly

as N

u

--3 m,

t h e r e e x i s t s an

depends only on

pJp0,

THEOREM 1.12.5.

where

and

E = [ [ a ~ b X] [x: llxll

5

V(t,XJX') _> ( t - a ) b ( [ l x ' I I ) ,

r

+m,

+

and

that ing

5

p] XRn],

on

[a,b],

Ilxl(t)ll _< N

on

[a,bl.

such t h a t

V

6

C[E,R+],

V(t,x,y)

< L i n the i n t e r i o r of

x E C(2'[[a,b],Rn]

p

Ilx(t)ll

This completes t h e proof.

(x,y), V ( a , x ( a ) , x ' ( a ) ) = 0, + + where b E C[R ,R ] with b ( r ) + m

D Vf(t,x,x')

Then f o r any solution

g.

Suppose t h a t t h e r e e x i s t s a

i s l o c a l l y Lipschitzian i n as

N

of (1.11.1)such

there e x i s t s an

80

E.

N >0

satisfy-

1 . 1 3 . EXISTENCE IN THE LARGE FOR SYSTEMS

Proof: such t h a t

By assumption on

b(r)

>L

if

r

b(r),

> N. -

t h e r e e x i s t s an

N

>0

A l s o , we have by t h e theory

of d i f f e r e n t i a l i n e q u a l i t i e s

v ( t , x ( t > J x ' ( t > )5 V ( a , x ( a ) , x ' ( a ) ) + L(t I f we now suppose t h a t f o r some

t2

E

- a),

t

(a,bl, Ilx(t2

then we a r e lead t o t h e contradiction

< (t2-a)[b(llx'(t2)11)-L] 5 V ( a , x ( a ) , x ' ( a ) )

0

= 0.

Hence t h e conclusion of the theorem i s t r u e . EXISTENCE IN THE LARGE FOR S

1.13

Y

S

~

We s h a l l f i r s t prove an existence r e s u l t analogous t o Theorem 1 . 5 . 1 f o r t h e BVP (1.13.1) where

f

XI'

E

= f(tJXJX')J

X(a) = A,

x(b) = B,

C[JxRnxRn,Rn], A,B E Rn.

THEOREM 1.13.1. Suppose t h a t t h e following conditions

hold: on J D-ai(t)

( i ) a , p E C[J,Rn] II C(l)[JoJRn] 0 and f o r t E J ,

such t h a t

a(t)

5

@(t)

_> f i ( t ~ a ( t ) ~ a ( t ~ i ) D-B'(t) )~ 5 fi(tJp(t)Jb(t>i))>

where

81

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

(iii)

hi

E

-I.

C[R ,(O,-)]

s a t i s f i e s (1.12.5).

and hi

Then f o r any .(a) 5 A 5 B(a), a(b) 5 B (1.13.1) has a solution x E C(2)[J,Rn] a(t)

5

vector

x(t)

No

p(b),

t h e BVP

such t h a t

5 p ( t ) on J . k r e o v e r t h e r e e x i s t s a constant

>0

Proof: -

5

such t h a t

Ix'(t)

I 5 No

on

J.

Ey hypothesis ( i i ) , we can choose a vector

depending only on c?,p,h,

as i n Theorem 1.12.1. L e t

N

> 0,

No be

such t h a t

Choose

c

> No.

Let

F(t,x,x')

be t h e modification function

as defined i n Definition 1.11.1r e l a t i v e t o the t r i p l e

a,p,c.

Then by Theorem 1.11.1it follows t h a t t h e BVP XI'

= F(t,x,x'),

has a solution

J.

x

x(a) = A,

x(b) = B,

with a ( t )

E C(2)[J,Rn]

-< x ( t ) -< B(t)

on

Hence we have X"

= F(t,x,x') = f ( t , x , x ' )

i n view of t h e d e f i n i t i o n of by ( i i )

,

5x 5

B(t), t

F.

Consequently, we obtain,

Recalling t h a t 1Z;l < ci and proceeding as i n the proof of meorern 1.12.1, it i s e a s i l y

whenever

shown t h a t

a(t)

( x '( t ) I

F(t,x,x') = f ( t , x , x ' )

E

J.

No on Jo.

This then implies t h a t

a d therefore

solution of t h e desired BVP

x(t)

is actually a

(1.13.1) completing t h e proof.

Based on Theorem 1.12.2, it i s possible t o exhibit.

82

1.13. EXISTENCE IN THE LARGE FOR SYSTEMS

another existence theorem which i s an extension of Theorem

1.5.5.

on JoJ where

83

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

SZ being the s e t

n = [(t,x,x): (t,x)

E

n

and cp(t,x)

,< x' ,< Jr(t,x)l.

-

Proof: Let F(t,x,xt) be a modified f'unction of as in Definition 1.11.2 f ( t , x , x l ) associated w i t h cu,@,cp,@ which i s continuous and bounded on J xRn xRn and therefore s a t i s f i e s -theses ( i ) and ( i i i ) of Theorem 1.13.1. Furthermore, in view of the assumptions relative t o a l , p l i n ( i i i ) , it i s easily verified that hypothesis (i) is true w i t h respect t o F ( t , x , x l ) i n place of f ( t , x , x ' ) . This implies that

condition (i) of Theorem 1.13.1 holds w i t h cp,@ instead of -c,c respectively. As a result, it follows, by Theorem 1.13.1 that there exists a solution x E: C(2)[J,Rn] t o the modified

BW x" = F(t,x,x'),

x(a) = A,

x(b) = B,

such that a ( t ) 5 x ( t ) 5 @ ( t ) on J, where @(a)= A = @(a), a ( b ) 5 B 5 @(b). We now apply Theorem 1.12.2 t o assert that x ( t ) is actually a solution of the BVP (1.13.1) satisfying ( t , x ( t ) , x ' ( t ) ) E 12, t E J. For t h i s , it is necessary t o check that a l l the hypotheses of Theorem 1.12.2 are s a t i s f i e d which we leave t o the reader since it i s similar t o the proof of Theorem 1.5.5. This completes the proof. EI(ERC1SE 1.13.1.

Under the assumptions of Exercises show that the BVP XI' = f ( t , x , x t ) , x(a) = A, x(b) '= B has a solution. 1.11.2 and Theorem 1.12.5,

1.14 FUR'MER RESULTS FOR SYSTEM Let us consider the d i f f e r e n t i a l system

(1.14.1) where f

X'l

E

= f(t,X,X'),

n n C [ [ O , ~ ] X R ~ X,R R

3,

84

subject t o the boundary

1.14. FURTHER RESULTS FOR SYSTEMS

conditions (1.14.2)

= 0, ~ ( 0 -AOx'(0) )

(1.14.3)

x(l)+A1x'(l) = 0 ,

AOjA1

being

matrices.

dxd

Here we s h a l l study t h e existence of solutions of t h e BVP (1.14.1)

- (1.14.3)

earlier results.

i n a more general s e t up than t h e

We employ mapunov-like fbnctions and t h e

theory of d i f f e r e n t i a l i n e q u a l i t i e s i n a s l i g h t l y d i f f e r e n t way which throws much l i g h t on t h e underlying ideas.

The

technique i s , of course, the modified function approach.

The

following lemma i s very u s e f u l i n our discussion. LEMMA 1.14.1.

(i)

U E

Assume t h a t

C ( * ) [ [ O J ~ ] J R + ] Jg

i s nonincreasing i n u

g(t,u,v)

c [ [ o J 1 X] R + X R J R - ] J

f o r each

-> g ( t J u J u ' ) ; ~ ' ( 1< ) 0, and

(1.14.4)

(t,v)

and

U''

(ii) some ct

E

u'(0)

2

0,

u(0)

-< m ' ( 0 )

for

-> 0 ;

(iii)

G E C [ [ O J ~ ] X R + ~ R ]and t h e r e exists an

such t h a t f o r u

2 L,

t

s a t i s f i e s t h e estimate

E

L>

[0,1],

r(t,TJO)

< aoJ t

E [oJT]J

where

a0 = min(+, 1/a); (iv)

t h e l e f t maximal solution

right minimal solution

(1.14.7)

p(t,O,O)

of

v ' = g(t,2L,v) 85

r(tJ1,O)

and t h e

o

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

e x i s t s on

[0,13.

Then there e x i s t s a

(1.14.8)

u(t)

5

Bo

>0

such that

and

Bo

5

lu'(t)l

Bo,

Proof: Assume t h a t the maximum of

0

u(t)

-< t -< 1.

occurs a t a

-

tl. From conditions u ' ( 0 ) > 0 and ~ ' ( 1 <) 0, it follows t h a t u'(t,) = 0 . Clearly tl > 0 , f o r otherwise

point

we would have

u(tl)

5 mt(tl)

>

Define

2L.

-

u ( t ) 10.

u ( t ) < 2L, 0 L for t E [O,tll.

W e s h a l l show t h a t

u(tl)

and consequently

= 0

If not, define

to = sup[t Since u(tl)

>

E

[O,t11:

U'(t)

to is well

by the mean value theorem,

2L,

defined.

It i s then e a s i l y seen t h a t

(1.14-9)

u'(to)

Setting

-> +u(t)].

2

L ,< u ( t ) ,

cuou(tO),

z(t) = u'(t)/u(t)

for t

t

[tO,tll

E

E

[tO,tll.

and using

assumption (1.14.5) we readily obtain ' ~ ' ( t 2) G ( t , z ( t ) ) , Notice t h a t

z(to) = 0

and

t

z(to)

E

[to,t,l.

1 cyo > 0 .

By the theory

of d i f f e r e n t i a l inequalities, we then i n f e r t h a t z(t) where with on

5

r(t,tl,z(tl)),

r(t,tl,z(tl)) T =

t

[tO,tl],

1'

t

E

[tO,tlI,

is t h e lef't maximal solution of (1.14.6)

Since z(tl) = 0, we see t h a t r(t,tl,O) < CY 0 and as a r e s u l t , we are lead t o t h e contradiction

This proves t h a t

u(t)

5

2L on

86

[0,11.

1.14. FURTHER RESULTS FOR SYSTEMS

U s i n g t h i s i n e q u a l i t y and t h e nonincreasing nature of g(tJuJv) i n

u,

we obtain

u"

-> g ( t , 2 L , u ' ) .

-> 0, ~'(1)5 0, d i f f e r e n t i a l i n e q u a l i t i e s , we have 0 p(tJ0JO)J

Again, using t h e f a c t t h a t theory of

where

for

and t h e

r ( t J 1 , 0 ) , p ( t J O J 0 ) are, respectively, l e f t maximal and

r i g h t minimal s o l u t i o n s of

on

u'(0)

[O,L]. 0

(1.14.7)which

Thus, we can f i n d a

-< t -< 1, B = max

[I

B

>0

a r e assumed t o e x i s t

such t h a t

Iu'(t)I
where max O< t <-l

r(tJIJo)lJ

I

min

- -

O < t < l

p(t,O,O)l].

The conclusion of t h e lemma now follows by choosing B = max[2LJB]. 0

The proof i s complete. 1

The functions g ( t , u , v ) = - k [ l + (2u)'+ 2 0, G(tjZ) = - ( a + k \ z I + Z ) J V(tjX) *11X112 are

COROLLARY 1.14.1. k

IVl]j

>

admissible i n Lemma 1.14.1, where h = $ a0e -(k+l),

provided

a

La

=

k(1

< aOke- (k+l)

+

(1/2h)

+ hk),

Our aim i s t o prove t h e following r e s u l t . IREOREM 1.14.1.

(a)

Assume t h a t

f E c[[OJ1]xRnxRnJRn]

and

A ~ , A a~r e p o s i t i v e

d e f i n i t e o r i d e n t i c a l l y zero; (b)

v

E

c ( 2 ) [ [ 0 , 1 ] XRnrR+],

g E C [ [ O J ~ ] X R + X R ~ R -g](~t , u , v )

each

( t J v ) ; and f o r

(t,X,X')

E

V(t,x)

i s positive definite,

i s nonincreasing i n [0,1] X R n X R n ,

(1.14.10) Vg(tJx) > g ( t J V ( t J x ) J V ' ( t J x ) )+ v / l f ( t J X J X ' )( 1 )

87

u for

u>oJ

1 . METHODS INVOLVING DIFFERENTIAL INEQUALITIES

where V'(t,x) = Vt(t,x) VF(t,x) = Vtt(t,x)

+ vx(t,x) x ' , + 2Vtx(t,x)

+ vx(t,x)

x ' + Vxx(t,X)X'

x'

f(t,x,x');

( c ) The boundary conditions (1.14.2), (1.14.3) imply, for

some a 2 0,

that

(1.14.12)

2 0, V'(l,x(l)) 5 V(O,X(O>) 5 (Yv'(O,x(O)); V'(O,x(O))

(d) G that f o r

u

E

and there e x i s t s an L

C[[O,l]R',R]

2 L,

t

E

7 E

and

>

0

such

[O,11

(l/u)g(t,u,v) and f o r any

0

-

(V/UI2

2 G(t,v/u)

(0,1], the l e f t maximal solution

r(t,7,0)

of Z'

= G(t,z),

Z(2)

= 0

s a t i s f i e s the inequality r(t,T,O) < ao, t E [O,T], where = min(+,l/a), (e) the l e f t m a x i m a l solution r ( t , l , O ) and the r i g h t minimal solution p(t,O,O) of a.

v ' = g(t,2L,v) exist on

[0,11.

Then there exists a solution x boundary value problem (1.14.1) Proof:

E

C(2)[[0,1],R

- (1.14.3).

Define the function

88

6(u,v)

dl

of the

on R+xR+

as

1.14. FURTHER RESULTS FOR SYSTEMS

osu, (1.14.1'3)

vLB,

( 1 i B - v),

O l u l B l v l B + 1,

(l+B-u)(l+B-v),

Blu,

0 5 v_
~ ( u , v )=

vlB+1,

otherwise. Next define t h e modified function

F

of

f

on

d d [0,1]X R X R

by

(1.14.14)

F(t,x,x'

Clearly t h e function [0,1] xRdxRd.

F

= S(llxll, IIX' Il)f(t,XjX'

1.

i s continuous and bounded on

Hence t h e r e e x i s t s a solution

x

E

C(*)[[O,l],R

d

]

of the boundary value problem

(1.14.15)

X"

= F(t,x,x'),

Set m(t) = V ( t , x ( t ) )

x(O)-Ax'(O) = 0 ,

X(1) + A I X ' ( l ) = O *

so t h a t , because of assumption ( c ) , we have

the r e l a t i o n s

which leads t o t h e f u r t h e r inequality

(1.14.18)

m"(t)

_> .g(t,m(t),m'(t)) + allx"(t)ll.

Hence, by Lemma 1.14.1,

it follows t h a t t h e r e exists a Bo > 0

such t h a t

89

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

As a result, s e t t i n g

-N = [min g(t,u,v): 0 ,< t 5 1, u ,< Bo, Ivl ,< Bol, we obtain from (1.14.18) m"(t) Thus, for 0

5

s

5

t

2 -N+ullx"(t)ll.

5 1,

2 IIx'(t)ll- IIx(l>ll- IIx(O)ll. Since V(t,x) i s assumed t o be positive definite, it follows, from the estimate V ( t , x ( t ) ) = m(t) 5 Bo, 0 5 t 5 1, t h a t (lx(t)II 5 B*, 0 5 t ,< 1. Consequently, we deduce that Ilx(t)ll

5 a*+

(2Bo +N)

O < t l l .

"B,

Eivdently, t h i s implies t h a t (1.14-19)

IIX(t)ll

€3

IIX'(t)ll

and

This, i n view of the definition of

5

B,

0

5 t 5 1,

assures us t h a t x ( t ) is actually a solution of t h e boundary value problem (1.14.)(1.14.3). The proof i s cmplete.

90

F,

1.14. FURTHER RESULTS FOR SYSTEMS

If f satisfies Nagumo's condition, assumption (1.14.10) may be weakened as the next theorem shows.

THEOREM 1.14.2. Let the hypotheses of Theorem 1.14.1 hold except that inequalities (1.14.10) and (1.14.U) are replaced by

2 g(t,V(t,x),V'(t,x))

(1.14.20) V;(t,x)

(1.14.21)

U(t,X,X')

+ u ~ ~ x ' ~ ~u,> 0,

+ 1 ,> IIX'II.

Suppose that Ilf(t,x,x')ll 5 h JJx'/Ifor (t,x,x') E [0,1] xRnxRn, and s ds/h(s) = 00. Then there where h E C[R+,(O,.o)] exists a solution x E C(2)[[0,1],Rn] of the boundary value problem (1.14.1) (1.14.3).

-

Proof:

We proceed exactly as in the proof of Theorem 1.14.1 until we arrive at inequalities (1.14.16). Consider first the case when u 11. Then, in view of (1.14.17), relations (1.14.20) and (1.14.21) yield the inequality (1.14.22)

Vi(t,x)

2 g(t,V(t,x),V'(t,x)) + (1-S,[IIx'II

-> g(t,V(t,x),V'(t,x))

z g(t,V(t,x),V'

+ u6I]x'II

- 13 -1+

IIx'~~[~(u1)+ 1 I

-

I(x'11,

(t,x>) 1

+

using the facts g 5 0, 0 5 S 5 1. Here we have used 6 for 6(u,v). If, on the other hand, 0 g(t,V(t,x),V'(t,x))

-> g(t,V(t,x),V'(t,x)) 91

- 1 + IIx'II[uS + (1-s)al

- 1 + u(Jx')(.

1. METHODS INVOLVING DIFFERENTIAL INEQUALITIES

Since by L a m a 1.14.1 we have

inequalities (1.14.22), (1.14.23) lead to m"(t)

2- ( N + 1 ) +

m"(t)

2- ( N + 1 ) + ollx'(t)ll,

IIx'(t)ll

and respectively, where, as before,

It then follows that 0 ( l ) From Lemma 1.12.1, we have IIX'(t>jl

5 (2B0 + N +l)/o

L r ( e W ) 5 ?(MI,

Letting B = m[B*,y(M)], the proof as before.

0

M

L

t

5

in any case. 1.

we obtain (1.14.19)which concludes

Finally, for later use, we shall state a uniqueness result leaving the proof as an exercise. THEOREM

1.14.3. Assume that f

V 6 C(2)[[0,1]xRnXRn,Rf], X = Y and

V(t,x,y)

E

C[[O,1] xRnxRn,Rn], 0 if and Only if

1.15, NOTES AND COMMENTS

Then the BVP (1.14.1), one solution.

x(0) = xo and x(1)

=

x has at most 1

1.15 NOTES AND COMMENTS For the existence results in the small contained in Section 1.1, see Hartman [31, Jackson [2], and Bailey, et al.

[31. The results of Sections 1.2 and 1.3 are taken from Jackson [2] except Theorems 1.2.4 and 1.3.2 which are based on the work of Schmitt [5]. Theorem 1.4.1, the Nagumo's condition, is taken from Jackson [2], while Theorems 1.4.2 and 1.4.3 are adapted from Schmitt [2] and Ako [I.], respectively (see also Hartman 133). Theorem 1.5.1 is taken from Jackson [21. Theorems 1.5.2, 1.5.4 are due to Erbe [l] and Theorem 1.5.5 is due to Schmitt [2]. The results of Section 1.6 are due to Bernfeld et al. [l]. Fxercise 1.6.1 contains the work of Schrader [2]. See George and Sutton [l] for the use of Wapunov-like functions. The contents of Sections 1.7-1.10 are taken from Jackson [2] where other related references may be found. The work contained in Sections 1.11-1.13 is due to Bernfeld et al. [41 except Lemma 1.12.1which is due to Lasota and Yorke [13]. The definition of the modified function as given in Definition 1.11.3 may be found in Hartman [31 where a number of results for second-order systems are given. The contents of Section 1.14 are due to Bernfeld et al. [j]. For further results in this direction, see Hartman [?I. For related results, see Knobloch [1,21, Schrader [5,61, Jackson and Schrader 111, Schmitt [31, Moyer [l], Gaines [3,41, Halikov [l], Gudkov and Lepin [l], Gudkov [21, and Mamedov [l].

93

Chapter 2 SHOOTING TYPE METHODS

2.0

INTRODUCTION

This chapter is essentially divided into two parts in which the shooting method serves as the underlying technique. The first part is concerned with the question as to whether uniqueness of solutions of boundary value problems implies the existence of solutions. Although the interdependence between uniqueness and existence is complicated for nonlinear equations, it can be formulated in simple terms for second-order differential equations. We examine this problem under linear and nonlinear boundary conditions and show that the results obtained are the best possible. The second part is devoted to an hportant method known as the "angular function technique." A number of results concerning existence, uniqueness, and criteria for existence of a finite OF infinite number of solutions are studied using this method as a tool. Einploying Lyapunov-like functions and the theory of differential inequalities, this technique is extended to cover nonlinear boundary conditions and systems of differential equations. 2.1

UNIQUENESS IMPLIES EXISTENCE

For linear differential equations, it is well known that the uniqueness of solutions implies the existence of solutions. For nonlinear equations the interdependence between uniqueness and existence is much more complicated. However, in the case of second-order differential equations this relationship can surprisingly be formulated in simple terms.

94

2.1. UNIQUENESS IMPLIES EXISTENCE

An extremely useful technique in handling these questions for second-order boundary value problems is the so-called shooting method. The idea of this method is to fix one initial value and to allow the slope at the initial point to vary through the real numbers. From the connectedness of the solution funnel and the uniqueness of BVP‘s, it is then possible to show that the values of the solutions at the final point cover the real line. In what follows we plan to illustrate this important technique in a detailed manner. Our discussion depends on the following variation of Kneser’s theorem. LEMMA 2.1.1. Consider the initial value problem X’ =

F(t,x),

x(to>

= X0’

where F E CCRXRn,Rn 1. Let S be any compact, connected set. Assume all solutions x(t,tO,xO) exist on [tO,tl]. Then

is a compact, connected set. Our first result will be concerned with a system of two first-order equations subject to simple boundary conditions. Consider the differential system x ’ = f(t,x)

(2.1.1)

with the boundary conditions Xl(tl)

(2.1.2)

where f result.

E

C [R x R ,R

Xl(t2)

=

2 2

= C2’

1. For this problem we have the following

95

2 , SHOOTING TYPE METHODS

0

J

THEOREM 2.1.1. =

(a,b).

Let J

=

(a,b],

-m

5 a < b < m, and

Assume

(i) fl(t,xl,x2) is an increasing function of x2 for fixed (t,x,) satisfying fl(t,xl,x2) + & m as x2 + k m uniformly on compact sets in (a,b)xR; (ii) all solutions of (2.1.1) exist on J; (iii) there exists at most one solution of (2.1.1), 0 (2.1.2) for all tl,t2 E J and all cl, c2. Then every BVP (2.1.1), (2.1.2) has exactly one solution 0 if tl E J , t2 E J. Proof: We may assume without loss of generality t2 = b. Let tl, c1 be given where a < tl < b. Let x(t) = be a solution of (2.1.1) satisfying the initial (x,(t),x,(t>) condition

Then define

S = [x(b): x(t) is a solution of (2.1.1) k satisfying (2.1.3) for each Irl 5 k]. By Lemma 2.1.1,

N

is connected. Let Sk be the projection of Sk on the x axis. Hence S is an interval which is 1 $ 0 nondecreasing in k. Define S = - Sk; then is an interval. To prove the stated result, it is sufficient to show S = (-m,m). If this is not true, suppose that S is bounded above. A similar proof holds if is bounded below. Then c (-m,M) for some M > 0 . Let xn(t) be a sequence of solutions of (2.1.1) satisfying Sk

-

N

z

&

N

N

s

(2.1.4) If s1,s2,

xn(tl)

...,sn

=

(cl,n),

n

=

,... .

1,2

is a sequence of t-values on a compact 96

2.1. UNIQUENESS IMPLIES EXISTENCE

f o r otherwise, from the well-known convergence theorems a subsequence of

[x ( t ) ]

n

converges uniformly t o a solution of

(2.1.1)J on a l l compact subsets of

J

0

,

a contradiction t o

(2.1.4). Let

x ( t ) be the f i r s t component of x n ( t ) . Observe n, 1 t h a t ~ ~ + ~ ,> x~n J( l (t t )) and x (s) < x (s) f o r n+l, 1 n, 1 t E (t,,6] and s E [ t l - 6 , t 1) f o r s u f f i c i e n t l y small 6. However, from ( i i i ) we immediately obtain

for

a < s < t l < t 0 and an i n t e r v a l [al,bll

such t h a t

From the mean value theorem

Ix ( t ) l 5 C f o r t E [a,,b,I. n, 1 there e x i s t s rn E [a J b ] such t h a t /xh,l(rn)( 52C/(bl- al). 1 1 Then condition ( i ) implies llxn(rn)ll i s bounded f o r all n. However, t h i s contradicts (2.1.5). of

[ t l - 6m,tl],[tl,tl+6m]

uniformly bounded. vn, wn

Letting

Hence, on any subintervals [x ( t ) l i s not n, 1 we see there e x i s t points

the sequence 6m + 0

where vn < tl,

v +t n 1’

wn>

w +t

n

1

such that, i n view of (2.1.6),

(2.1.7) as

n

x

n,l

(v ) n

+-m

and

xn,l(wn>

++a0

+m.

Let x*(t)

be any solution of (2.1.1) satisfying

91

2. SHOOTING TYPE METHODS

x*(b) = M; hence, since S E (-m,M),xT(b) 1 n. From (2.l.7), f o r l a r g e n

x ( t ) and xT(t) n, 1

This implies

d i c t i r q assumption ( i i i ) .

> xn,l(b)

for a l l

i n t e r s e c t twice, contra-

This completes the proof of Theorem

2.1.1. COROLLARY 2.1.1.

where

Consider the BVP

h E C[JxRxR,R], J = (a,b]. (i)

Assume t h a t

a l l solutions of (2.1.8) e x i s t on

(a,b];

( i i ) t h e r e e x i s t s a t most one solution of (2.1.8), (2.1.9) f o r tl,t2 E (a,b], c1,c2 E R. Then there e x i s t s exactly one solution of (2.1.8) and (2.1.9). Proof:

The BVP (2.1.8),

xi

= x2,

(2.1.9) can be w r i t t e n as = h ( t , x ,X

X'

1 21,

2

x (t ) = C2' 1 2

x ( t ) = c 1 1 1'

We see immediately t h a t t h e conditions i n Theorem 2.1.1 a r e satisfied.

Hence Corollary 2.1.1 follows.

EXERCISE 2.1.1.

suppose t h a t

Consider t h e BVP (2.1.8),

h(t,x,x')

is s t r i c t l y increasing i n x.

Assume that a l l solutions of (2.1.8) exist on there e x i s t s exactly one solution of (2.1.8),

tl,t2

E

(a,b]

Hint:

and a l l

(2.1.9) and

c1,c2

E

(a,b].

Then

(2.1.9) f o r a l l

R.

Show t h a t there exists a t most one solution by 98

2.1. UNIQUENESS IMPLIES EXISTENCE

showing t h a t t h e difference of any two solutions cannot a t t a i n a positive maximum o r negative minimum.

Then use Corollary

2.1.1. EXERCISE 2.1.2.

Assume

h(t,x,x')

satisfies

where

Then (2.1.8),

Hint:

(2.1.9) has a unique solution. Show t h a t t h e inequality

q(t)ixl + p(t)lx'I

lx"I

with boundary conditions x ( t l ) = 0, x(t,) = 0 has only the zero solution, where I t2- t I C b a. Then use Corollary 1 2.1.1.

-

Observe t h a t i n Theorem 2.1.1we obtained t h e existence of a solution on

(a,b].

t h a t we cannot s u b s t i t u t e EXAMPLE 2.1.1.

I n t h e following example we show [a,bI

for

(a,b].

The implicit equation

P > -2,

cp + (P/2) t a n - l cp = 9,

The family of a l l solutions of

has a unique solution cp(p,q). (2.1.10)

x'1 = - x +

1

3

tan-'

cp(sin t , x sin t

+

x ' cos t )

can be represented by x ( t ) = A cos t where A

and

B

+

B sin t

+ 3 tan-1 B,

a r e a r b i t r a r y constants.

Consider the

boundary conditions ( 2 . 1 . ~ ) x ( t l ) =xl

and

x ( t2 ) = x2, 99

tl,t2

E

[o,~].

2 . SHOOTING TYPE METHODS

Then x i

=

A cos ti + B sin ti +

Eliminating A, (2.1.12)

1

2

tan-1 B,

i

=

1,2.

we obtain

-

B sin(t2 t ) + ~ ( C O Stl - cos t2) tan-1 B 1 = x2 cos tl- x cos t2' 1

We now claim that there exists at most one solution of (2.1.10), (2.1.11). If 0 5 t l < t2 < T or 0 < t < t2 5 T , then 1

sin(t2- tl) > 0

cos t2- cos t < 0 . 1

and

Thus from (2.1.12), we see that, for any xl, x2, tl, t2, B and A are uniquely determined. Thus solutions of (2.1.10), (2.1.11) are uniquely determined. For the case t = 0, t2 = 7, (2.1.12) reduces to 1 (2.1.13)

tan-'

B

=

x1 + x2'

Once again uniqueness of solutions foUows. Thus on [ O , T ] the conditions of Corollary 2.1.1 hold. However, observe (2.1.13) implies there exists no solution, if )x1+x21 = T/2EXERCISE 2.1.3.

(2.1.14)

Show that solutions of y"

=

-y + arctan y,

(2.1.15)

when they exist, are unique for t ,t E [ 0 , 7 ] . Then show 1 2 there exists no solution of (2.1.14) satisfying (2.1.16)

Hint:

Y(0)

= 0,

Y(T)

=

3T.

Assume there exist two solutions y,(t),

100

y,(t)

of

2.2, GENERAL LINEAR BOUNDARY CONDITIONS

(2.1.14), y,(t)

tlyt2 E

(2.1.15) f o r

- y2(t)

satisfies

i s a subfunction of subinterval of W(T) > u(T),

w"(t)

y" = -y. ul' = -u

u ' ( ~ / 2 ) . Prove that

u(t)

leads t o a contradiction. of (2.1.15), (2.1.16), vl' = -v

and

Argue .that y(T) y(O) = 0 , ~ ' ( 0 )= m.

m.

and

x'.

> u(O),

has two zeros on

[O,T]

which

To show t h e r e exists no solution

~ ' ( 0 )= m

+ 1,

of

[O,T]

with respect t o (2.1.14) f o r any

< v(T)

= 2T,

where

y

s a t i s f i e s (2.1.14),

I n t e r e s t i n g l y enough we can, under c e r t a i n derive t h e r e s u l t s of Corollary

2.1.1 f o r t h e case when J = [a,b]. h

i s not a proper

w(0)

v(t)

conditions on h ( t , x , x ' ) , example, t h a t

w(t)

w ( ~ / 2 )= u ( T / ~ ) , w ' ( T / ~ )=

v(0) = 0 ,

i s a subfunction on

and prove that

[tlyt2]

If

=

w(t)

prove t h a t t h e solution

+ T,

REMARK 2.1.1.

2 -w(t)

then show that

[O,T], where

Show t h a t

(0,T).

It i s s u f f i c i e n t , f o r

s a t i s f y a Lipschitz condition i n both x

This follows by considering t h e l i n e a r equation

derived from t h e Lipschitz condition, and using t h e f a c t t h a t t h e uniqueness of t h e BVP f o r t h e l i n e a r equation on a l l subi n t e r v a l s of

implies t h e existence of unique solutions

[a,bl

on a l l subintervals of

of BVP's f o r x" = h ( t , x , x ' ) Moreover, an estimate of

[a,b]

[a,b].

can be derived i n terms of

hyperbolic f'unctions. 2.2

GENERAL LINEAR BOUNDARY CONDITIONS "he question as t o whether uniqueness of solutions of

(2.1.8) with t h e general l i n e a r boundary conditions (2.2.1)

alx(tl)

+

ap'(tl)

= cl,

blx(t2)

+ b p ' ( t 2 ) = c2

implies existence i s n a t u r a l l y more d i f f i c u l t since it depends upon t h e c o e f f i c i e n t s

al,

a2, bly b2.

101

2. SHOOTING TYPE METHODS

We shall show, as a consequence of our next result, that Corollary 2.1.1 holds for (2.1.8) and (2.2.1) whenever a2b2 = 0 . THEOREM 2.2.1. Let J, x, f be as in Theorem 2.1.1 except that condition (iii) is replaced by the following:

(iv) there is at most one solution of (2.1.1) satisfying (2.2.2)

x1(t1)

for any tl,t2

E

=

J

0

c1

and

x2(t2)

and c1,c2

E

=

c2'

tl

#

t2'

R.

Then (a) every boundary value problem (2.1.1), (2.1.2) has a unique solution; (b) every boundary value problem (2.1.1), (2.2.2) has a unique solution. Proof: In order to show (a), we must show condition (iii) of Theorem 2.1.1 holds. Let x(t), y(t) be distinct solutions 0 of (2.1.1) satisfying xl(tl) = y (t ) for some tl E J It 1 1 0 suffices to show that xl(t) # yl(t) for tl f t E J We show this for tl < t x2(t) for t E (tl,b). Then either

.

.

Y2(t1> = X2(tl) or Y2(t1> > X2(tl)' If Y (t 1 = X2(tl)' 01 then, by (iv), yl(t) f x,(t) for tl # t E J If then yi(t,) > xi(tl) by (i), so that there y2(tl) > x2(tl), exists a 6 > 0 such that yl(t) > x,(t) for t E (tl,tl+6). Assume that there exists a least T,tl< 7 < b such that xl(T) = yl(T). However, then ~'(7) 2 y'1(T), which implies 1 x2(T) 2 y2(T). This contradiction shows that xl(t) # yl(t) on (tl,b) and thus completes the proof of part (a).

.

102

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

The proof of part (b) i s similar t o t h a t of Theorem 2.1.1.

t2 =

Assume without l o s s of generality t h a t Define

Sk a s i n Theorem 2.1.1 and l e t

projection of

3

on t h e

Sk

i s an i n t e r v a l . A

bounded below).

3

axis.

Let

It suffices t o show

s^

3 =U,

m

be t h e r

z

but

$

(--,MI

2.

then

S,;

i s the r e a l l i n e .

We may assume t h a t t h e r e e x i s t s

(-m,M)

C

x2

8,

i s bounded above ( a similar proof holds i f

Suppose S that

b.

M

A s before, l e t

^s

>0

is such

xn(t)

be

a sequence of solutions of (2.1.1) s a t i s f y i n g

n

xn(tl) = (c,,n),

(2.2.3)

= 1,2

,... .

Then there e x i s t sequences

such t h a t (2.1.7) holds vn, "n [ t h i s follows since ( a ) i s the same as ( i i i ) i n Theorem 2.1.11.

From (2.2.3) there e x i s t s a sequence {zn] such t h a t t
+ t l and xn2(zn) 1 n solution of (2.1.1) s a t i s f y i n g

as

n

403.

x*,(b) = M.

Let

x*(t)

be any

From t h e f a c t t h a t

^s c (-m,M), xz(b) > xn2(b) f o r a l l n, but f o r l a r g e x:(zn) < xn2(zn). Thus there exists zk such t h a t

n

(2.2.4)

From (2.1.7)

< x,(wn),

x;(wn)

so t h a t t h e r e e x i s t s

zk

1

#

d',

> xnl(vn)

n

v n

n

and

n

< znt t < wn.

then (2.2.4) and (2.2.5) contradict ( i v ) .

Suppose z ' = z t t - then define x ( t ) = x n ( t )

n

for large

such t h a t

z"

X*(Zt') = Xd(ZB),

(2-2-5) If

xT(vn)

n'

x ( t ) = x*(t)

for

of (2.1.1)

such t h a t

x*,(b) = M,

so t h a t

z'

n

Gl(tl)

(--,MI

< t 5 b. = xnl(tl) E

3,

for

Then x ( t ) = c1

a

zk,

i s a solution

and x2(b) =

a contradiction.

This

completes t h e proof of Theorem 2.2.2. COROLLARY 2.2.1.

Consider t h e BVP (2.1.8),

103

(2.2.1) and

2 . SHOOTING TYPE METHODS

assume

a2b2 = 0.

Suppose a l l s o l u t i o n s of (2.1.8)

e x i s t on

( a , b ] and t h e r e e x i s t s a t most one s o l u t i o n o f (2.1.8), (2.2.1) f o r every tl,t2 E ( a , b ) . Then t h e r e e x i s t s

b

(a)

a unique s o l u t i o n o f (2.1.8),

(2.l.g),

(b)

a unique s o l u t i o n o f (2.1.8),

(2.2.1).

Proof: -

2

= 1 and

Without loss of g e n e r a l i t y , assume a1 = 1. Then (2.1.8),

a s t h e f i r s t - o r d e r BVP f o r X'

1

L-

x2-b x 1 1'

X$ =

x

=

(2.2.1)

(xl,x;

h(t,Xl,x2

+

a2 = 0,

can be w r i t t e n

blxl)

- b 1x 1)

and

i n which

+ bl(X2

- blxl),

We t h e r e f o r e see t h a t t h e hypotheses o f Theorem 2 . 2 . 1 h o l d and t h e r e s u l t follows. Assume i n (2.1.8) t h a t t h e f u n c t i o n

EXERCISE 2.2.1. h(t,x,y) lhxI 5 M, for

E

2

C(l) [ (a,b) xR , R ] s a t i s f i e s t h e i n e q u a l i t i e s IhyI 5 K. If I b - a l < l y / b ((u 2 +Klul +M)-')du

al = 1, a2 = 0, bl = 1, b2

#

0,

2

then t h e BVP (2.1,8),

(2.2.1) has a unique s o l u t i o n .

Hint: -

Prove (2.1.8), (2.2.1) has a t most one s o l u t i o n .

!Phis can be done by assuming t h a t t h e r e e x i s t s o l u t i o n s x2

a n d b y s e t t i n g x = x1-x2

obtain

xl,

lx"I < Mix1 + Klx'I,

x ( t ) = 0, x ( t 2 ) + b 2 x T ( t 2 )= 0. Then l e t t i n g w = x ' / x , 1 < w2 + KIWI + M, w ( t l - 0 ) = + m and w(t 2 ) = -l/b2. deduce ( w ' 1 By t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s conclude t h a t

w(t) < u ( t ) ,

where

-u' = u2

u(t)

+

i s the solution of

Klul + M,

F i n a l l y o b t a i n a contradiction. 104

u(t2) =

- l/b2.

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

I n t h e next two examples, we show t h a t Corollary 2.2.1 does not necessarily hold without t h e assumption t h a t a2b2 = 0 . EXAMPLFi 2.2.1.

(2.2.6)

The equation

2t l + t2

=

2 - x, 1+t2

X'

has the general s o l u t i o n x ( t ) = A(t2

-m

- 1) + B t .

+m

Hence there

e x i s t s one solution s a t i s f y i n g t h e boundary conditions x l ( t , ) = c1,x1(t2) = c2.

Observe

x 1( t )

are two d i f f e r e n t solutions s a t i s f y i n g

and

0

x2 ( t )

p

x(-1) = 0, x(1) = 0.

Moreover, t h e r e e x i s t s no solution s a t i s f y i n g

x(-1) = 0,

x(1) = 1. Thus (a) i n Corollary 2.2.1 i s violated.

EXAMPLE 2.2.2.

Let

be defined as i n Example 2.1.1.

Cp

Consider t h e d i f f e r e n t i a l equation

with t h e boundary conditions xt(t,)

- x(tl)

x ' ( t 2 ) - x ( t 2 ) = r 2'

= r 1'

The family of solutions of (*) i s given by x(t) = A where

A

and

B

+

Bet + t tan-'

B,

a r e a r b i t r a r y constants.

conditions lead t o ( l - t i ) tan Eliminating A

-1

B-A=

we obtain (tl

- t2)

tan

-1

105

B = r

2-rl

The boundary

-

t2 1

2 . SHOOTING TYPE METHODS

and thus t h e r e e x i s t s a t most one s o l u t i o n of t h e BVP.

How-

ever, i f

-

Ir2 rll _> (7J/2) It2-t1I, t h e r e e x i s t s no s o l u t i o n .

Hence (b) i n Corollary 2.2.1 does

not hold. When (2.2.7)

a2b2

#

rx(tl)

we can r e w r i t e (2.2.1) i n t h e form

0

+

x l ( t ) = c 1’ 1

s x ( t 2 ) + x l ( t2 ) = c

2‘

We now present a r e s u l t i n which uniqueness of (2.1.8) and (2.2.7)

implies existence under c e r t a i n added conditions.

-

Consider t h e ordering on

(al,bl)

Write

I: (a0,b0) (al,bl)

defined a s

R2

a. < bo

and

a1

5

a.

and

bl

3 bo,

> bo

and

a

2

a.

and

b

5

a.

1

1

bo.

< (aO,bo) when (al,bl) 5 (aO,bo) and

(al,bl) # (aO,bo). noted by ( h ; r , s ) . THEOREM 2.2.2.

Problem (2.1.8) and (2.2.7) w i l l be de-

Assume a l l s o l u t i o n s of (2.1.8) e x i s t on

(a,b) and a r e uniquely determined by t h e i r i n i t i a l values. Assume problem ( h ; r o , s O ) i s unique f o r any tl,t2 E (a,b), then t h e r e e x i s t s a unique s o l u t i o n o f

(h;rl,sl)

whenever

(rl,sl) < (r0,so)* Proof:

Assume t h a t

The o t h e r cases have s i m i l a r arguments and t h e r e f o r e we s h a l l consider t h i s case only. x(t,v)

Fix

tl, t2, and

c1

be t h e s o l u t i o n of (2.1.8) s a t i s f y i n g

106

and l e t

2.2. GENERAL LINEAR BOUNDARY CONDITIONS

x ( t 1, v ) = v,

x'(t,,v)

= cl-r

+

soX(t,v).

0

v.

Define T(t,v) = x ' ( t , v )

(h;ro,sO), v + T (t , v )

From t h e uniqueness o f mapping f o r

t

1 tl.

Define

( T(t,v2)

AT/Av Then

AT/&

t'tl f o r a l l t 2 tl.

=

i s a one-to-one

-ro + so

>

- T(t,vl))/(v2 - vl). 0.

Thus

+

so h / A v

AT/Av

i s positive

'ken, s i n c e

AT/Av = h ' / A v

>

0,

we see t h a t

Setting S(t,V) = x ' ( t , V ) + SIX(t,V) = T(t,V)

Hence

v +S(t2,v)

+

(sl

- S,)X(t,V),

i s a one-to-one mapping of

This i s equivalent t o

(h;rl,sl)

R

onto i t s e l f .

having exactly one s o l u t i o n

and thus t h e proof i s complete.

i s s t r i c t l y increasing and t h a t s o l u t i o n s of (2.1.8) e x i s t on (a,b) and a r e

COROLLARY 2.2.2. in

x

Assume

h(t,x,y)

uniquely determined by i n i t i a l conditions.

s

> 0,

problem (2.1.8),

Then, f o r

(2.2.7) has a unique solution. 107

r < 0,

2 . SHOOTING TYPE METHODS

Proof:

ro, so

To prove t h i s , choose

r < ro < 0 , 0 < so < s .

Hence

(r0,so) > (r,s) and by it s u f f i c e s t o show (h;ro,sO) i s unique.

Theorem 2.2.2,

Assume t h e r e e x i s t two s o l u t i o n s x = x1-x2.

and x2 and l e t 1 attains either its

x ( t ) f 0, x ( t )

Since

x

p o s i t i v e maximum o r negative minhum on x

such t h a t

Assume t h a t

[tl,t2].

a t t a i n s i t s p o s i t i v e maximum (the case i n which

i t s negative minimum uses a similar argument).

tl,

occurs a t

then

maximum occurs at

t h e second condition

s x(t ) 0

to

t h e maximum occurs at

>

x'(tl)

5

If t h e maximum

and t h e f i r s t

0,

r o x ( t ) + x ( ) = 0 cannot hold. If t h e 1 ' t2, then x ( t 2 ) > 0, x ' ( t 2 ) 2 0, and

boundary condition

x(t,)

> 0,

x(tl)

x attains

2

E

+

xt(t,) = 0

(tl,t2),

x ' ( t o ) = 0,

0,

however, by t h e monotonicity o f

cannot hold.

If

then x"(to)

h(t,x,x'),

5

0;

we have

a contradiction.

I n t h e next e x e r c i s e , we show t h a t t h e s t r i c t i n e q u a l i t y (rl,sl)

< (ro,so) cannot be weakened t o t h e i n e q u a l i t y

(rlYs1) 5 (ro,so). This w i l l provide another example i n which a2b2 # 0 and uniqueness o f t h e boundary value problem does not imply existence. EXERCISE 2.2.2.

Construct an example i n which t h e

hypotheses of Theorem 2.2.2 hold with replaced by t h e weaker condition

( y l ) < (ro'so) (rl,sl) 5 (ro,so) such

t h a t t h e r e e x i s t s no s o l u t i o n of t h e problem (2.1.8), (2.2.7).

Hint:

Let

Cp(p,q)

CP + p a r c t a n CP = q.

be t h e s o l u t i o n of t h e equation

Consider t h e d i f f e r e n t i a l equation 108

2.3. WEAKER UNIQUENESS CONDITIONS XI'

= F"cp(l/F',

x t / F t ) on t h e i n t e r v a l

w i l l be determined.

Let

w = x';

(-1,2),

w

then

where

F

s a t i s f i e s the

general solution

+

w ' ( t ) = BF'(t)

Solve f o r

x(t)

i n terms of

w(t)

conditions, where we may assume conditions on

F(tl),

This leads t o

1 where

F2

+

-r ( t + l )

1. Then obtain

t

F'(t)

>0

E

[-LO],

6

[0,11,

E

[1,21,

on

satisfies

tl

=

I (rc2 - c1)/(2r

-

F2(0) = 1 e'r, g(0) = l / r t g(1) = 1, g ' ( 0 ) = g'(1) = 0. Show

such t h a t

-4,t2 = 3/2 - 1)I 2 T / 2 ,

-< 1,

and

and f o r cl, c2 chosen so t h a t t h e r e e x i s t s no solution o f

(Use ideas i n B m p l e 2.1.1,)

(2.1.8) and (2.2.7). 2.3

>

t t

( ~ ~ (- 1l)el-t, )

i n which g 6 C ( l ) [ [ O , l ] , R ] l/r < g ( t ) < 1 for 0
tha t for

s = 1, r

and such t h a t

F2 + g(t)F; = 1,

-

and s u b s t i t u t e t h e boundary

F f ( t l ) , F ( t 2 ) , F 1 ( t 2 ) such t h a t unique-

ness holds (see Example 2.1.1) (-1,2).

a r c t a n B.

WEAKER UNIQUENESS CONDITIONS

It i s n a t u r a l t o ask whether it i s possible t o weaken t h e uniqueness condition of t h e BVP (2.1.8), (2.1.2) by not demanding t h a t t h i s condition hold f o r a l l tl,t2 [a,b+&) (or

tl,t2

E

(a-&,b]).

6

I n t h i s section, we plan

t o discuss how some of t h e previous r e s u l t s remain t r u e even when t h e uniqueness assumptions a r e weakened f o r and f o r

t2

E (b

- &,b + E )

THMlREM 2.3.1.

Let

f o r some h

6

E

>

0.

C[[a,b+E) xR2,R]

109

t1 = a f o r some

2 . SHOOTING TYPE METHODS

>

E

and suppose t h a t a l l s o l u t i o n s of i n i t i a l value problems

0

[a,b + E).

f o r (2.1.1) e x i s t on

Assume t h a t f o r a f i x e d

t h e r e e x i s t s a t most one s o l u t i o n on

[a,t2]

c

o f t h e BVP

1

(2.1.1) and (2.3.1)

.(a)

for a l l

= c1 '

and t2

c2 E R

x ( t 2 ) = C2'

(b

E

- E,

b + E).

Then t h e r e e x i s t s

e x a c t l y one s o l u t i o n of (2.1.1) and (2.3 - 2 )

.(a)

f o r each

= cl,

x ( b ) = c2

c2 E R. For each m E R

Proof:

define

Am = [x(t,m) E C ( 2 ) [[a,b

+E)

x R , R ] : x(t,m)

i s a s o l u t i o n of (2.1.1) with

x(a) = c

x ' ( a ) = m].

and Define t h e mapping

r(m) = [x(b,m)]; I?

maps p o i n t s i n t o s e t s

and an a p p l i c a t i o n of Lemma 2.1.1 y i e l d s t h a t connected s e t s i n t o compact, connected s e t s . then

i

5

r(m)

m <m2 < m 1

r(mi).

m2 < m

t h e BVP.

3

Since

there exists

m*

ml

#

m2,

5 max r(m2).

We now show E

maps compact,

fl r(m2) = $, because of t h e uniqueness of s o l u t i o n s We can thus define r(m,) 5 r(m2) i f and only i f

max r(m,)

x

r

If

r(ml)

of BVP.

that

1

m*

3'

E

i s monotone.

Assume, f o r example,

r(m,) < r(m3) < r(m2). Let < x < x2 and F ( [m ,m 1 ) i s connected, 1 3 1 2 [m,,m2] such t h a t x(b,m+) = x Since 3' and

x

we have a c o n t r a d i c t i o n t o t h e uniqueness of

The o t h e r cases may be t r e a t e d by a similar argument.

I n order t o complete t h e proof of Theorem 2.3.1, enough t o show

r ( R ) = R.

If not, t h e r e exists a

9

We f i r s t prove t h a t E

r(R) 110

r(R)

it i s i s open.

t h a t i s not i n t h e i n t e r i o r .

2.3. WEAKER UNIQUENESS CONDITIONS

Hence there i s an

r

t o n i c i t y of

r ((;,-GI)

m

such t h a t

x(b,m) = JI.

we can find m > m N

contains

-

< m such t h a t

or

as an i n t e r i o r point.

JI

contradict ion.

r(R)

It thus suffices t o prove t h a t

above nor below,

By the mono-

r(R)

Assume

This i s a

i s neither bounded

i s bounded above (a similar

argument i s t r u e f o r the other case) and l e t '1 = sup r(R). L e t xn(t,m ) denote a sequence of solutions of (2.1.1), n (xn(a,mn)) = cl,(x;(a,mn)) = mn such t h a t xn(b,mn) + q . For c E ( b , b + & ) and from the uniqueness of BW,

-

xn(c,mn> xn(b,mn) C-b

where

K = min {0, (x1(c,m,)

> -

- xn(b,mn)

xl(c,ml)

-> K

C-b

- q)/(c - b)].

We m a y assume t h a t , f o r i n f i n i t e l y many nls, we have x;(b,mn) 5 0, because a similar argument can be made f o r c E (b &,b) i n case x;(b,mn) _> 0 f o r i n f i n i t e l y m a n y n ' s .

-

We r e s t r i c t our arguments t o those Let

sn

=

[t:

o> x;(t,mn) 2

n

such t h a t

K, t E [ b , c ~ ] .

This s e t i s nonempty by the mean value theorem.

-

s = min S then b < s n < c and n n' without loss o f generality, sn + s

x;(b,mn)

If

x;(b,mn) x;(t*,mn)

t*

E

<

= 0,

0,

= 0

then

b = s

n then xn(t,mn) < q

x;(b,mn)
0 and.

kt

- n n n 5

K < x t ( s ,m )

0.

Thus,

(s m ) +?,. n n' n xn(sn,mn) < q. If

and

XI

t E [b,t*l, where and t* i s the f i r s t such point. men

Sn and therefore

for

sn E [b,t*].

Thus again xn(sn,mn) < q . [b,c]] (from the

Since xn(sn,mn) _> min [xl(t,mn): t E uniqueness of BVP), there e x i s t s a subsequence of

{xn(sn,mn)] which we choose t o c a l l by the same labeling, such t h a t

-

x (s ,m ) +xo. Thus, sn + s o , x (s ,m ) + x o , x;(sn,mn) + x o . n n n n n n Let S(6) be a 6 - b a l l i n R3 about ( S ~ , X ~ , ~ ~ ) , 111

2. SHOOTING TYPE METHODS

where

6

- so.

E

For

n

s u f f i c i e n t l y large

(snJxn(~nJmn),x~(sn,~n)) E ~(6). Since a l l solutions of (2.1.8) e x i s t on Ix$(t,m,)I

[a,b

there exists

+&),

< Q for t

E

[ a , s O + 6 ] . Let

be solutions o f (2.1.8) with montonicity of X(b9-Q)

2

q

or

x(b,*), x(b,Q)

Q

x(b,-Q)

> Xn(bjmn)

such t h a t

x(t,Q)

x(a,+Q) = cl;

either

>0

and

x(t,-Q)

then by the

> xn(b,mn)

which means

so that

x(b,Q)

2

q.

This i s a contradiction and t h e proof of Theorem 2 . 3 . 1 i s

complete. COROLLARY 2.3.1.

the BVP (2.1.8),

Under the conditions of Theorem 2.3.1,

(2.2.1) has a unique solution, i f

We leave the proof as an exercise.

m a y be helpful.

For a fixed

a2b2 = 0.

The following h i n t s

Consider the boundary conditions

clJ

define

r(m) = [x(b,m): where

x(t,m)

of (2.1.8) such t h a t

x' (.a,m)

= c1

- rlml.

i s a solution

x(a,m) = m

Then using the same techniques as i n Theorem 2.3.1,

and

show

r(R) = R.

By combining the techniques employed i n Theorems 2.1.1 and 2.3.1,

one can deduce t h e following generalization, the

proof of which we leave t o the reader. THEOREM 2.3.2.

Consider t h e BVP (2,1.1),

(2.1.2)

assuming conditions ( i ) and ( i i ) of Theorem 2.1.1.

Suppose

t h a t t h e condition r e l a t i v e t o t h e uniqueness of BVP, i n the sense of Theorem 2.3.1 i s t r u e . solution of (2.1.1),

(2.1.2).

112

Then t h e r e e x i s t s a unique

2.4. NONLINEAR BOUNDARY CONDITIONS

2.4

NONLINEAR BOUNDARY CONDITIONS We now present a r e s u l t i n which t h e boundary conditions

are. nonlinear.

Under some additional hypotheses on h ( t , x , x ‘ ) ,

it i s possible t o deduce existence r e s u l t s f o r general l i n e a r boundary conditions f o r which corollary t o our main r e s u l t . THEOREM 2.4.1.

S1, let

as

x2

-L

g2 (c)

+ y2

--tm

i n e i t h e r direction.

R2

be nonempty.

Assume f o r

t

uniformly f o r

E

(b

- E,b + E)

and a l l

one solution of (2.4.1)

Proof:

f = (fl,f2)

E [a,b+&),

and t h a t a l l

(2.4.2) c E R,

- (2.4.3).

and

g,(x(s),y(s))

= c

there e x i s t s exactly

The idea of the proof i s similar t o t h a t of the

proof of Theorem 2.3.1. gl(x,y) = c1

For every

[ a , b + & ) . men i f there e x i s t s

a t most one solution of (2.4.1),

s

This we s t a t e as a

Y ’ = f2(t,X,Y),

solutions of (2.4.1) e x i s t on for

0.

when parameterized i n terms of a r c length, leaves

every bounded region i n C,

#

Consider t h e boundary value problem

x’ = fl(t,X,Y),

(2.4.1)

then

a2b2

Let

Po

and parameterize 113

be an a r b i t r a r y point of

S1 by a r c length with

Po

2 . SHOOTING TYPE METHODS

a s t h e i n i t i a l point.

The points on

s

For any f i x e d

t h e parameter.

E

S1 w i l l be denoted by

[a,b + E ) ,

define f o r

s1

P E

Ts(P) =

[ (x(S,P),Y(s,P)),

where

(x(t,P),y(t~P))

denotes a s o l u t i o n of (2.4.1) s a t i s f y i n g ( x ( a , p > , y ( a , p > ) = PI, and l e t

rs(p) =

[g2(x(s,p),y(s,p)):

( X ( S , P > ~ Y ( S , P >E ) T ~ ( P > ] .

It i s s u f f i c i e n t t o show rb(R) = R, t o prove t h e theorem. From Lemma 2.1.1, Ts(A) i s compact and connected, i f A

i s compact and connected.

g (x,y) i s continuous, 2 It follows from a simple

Since

i s compact and connected.

rs(A)

argument t h a t

rs(R)

i s connected. rs(P1) n rs(P2) = @ f o r i s an i n t e r v a l o r p o i n t . We

By uniqueness o f t h e BVP, s E (b

- E,b + E ) .

rs

now show t h a t

Thus

r S (P)

i s monotone f o r

s

(b - E,b

E

+ E).

Let

< P2 < P (ordered by a r c l e n g t h ) and assume rs(P2) < 3 rs(P 1) < r s ( P 3 ) ( t h e o t h e r cases follow s i m i l a r l y ) . Let

P1 x

E rs(Pi), i = 1,2,3. Since i t h e r e e x i s t s P*, P2 5 P* < P

Since

P

1

< P2 5 P*,

i s connected,

rs([P2,P3])

E rs(P*). 1 t h i s c o n t r a d i c t s uniqueness o f s o l u t i o n s

- 3

such t h a t

x

of boundary value problems. Assume now

rb(R)

i s not a l l of

R

and t h a t

bounded above [ a s i m i l a r argument holds i f below], that

q

Let E

q = sup rb(R).

rb(R).

(x(t),y(t))

We claim

rb(R)

q d, rb(R).

is

rb(R)

i s bounded Suppose

Then t h e r e e x i s t s a s o l u t i o n of (2.4.1),

s a t i s f y i n g (2.4.2) and g2(x(b),y(b)) = q.

Let

and choose P2, P3 such t h a t P2 < P1 rb(P1)

P1 = ( x ( a ) , y ( a ) )

or

rb(P3)

>

rb(Pl),

Since

q E rb(Pl)

114

t h i s contradicts the

2.4. NONLINEAR BOUNDARY CONDITIONS

maximality of

q k, rb(R).

and thus

q

There e x i s t s

zn

rb(R)

E

such t h a t

zn t q.

To each

t h e r e e x i s t s a unique point P E S1 and a unique s o l u t i o n n n of (2.4.1), ( x n ( t ) , y n ( t ) ) through Pn a t t = a such t h a t z

g2(xn(b),yn(b)) = zn.

Also

qn(t) = g2(xn(t),~,(t)), t

Pn E

i s monotonic i n

n.

Let

[a,b +&I. Then

A s i n t h e proof of Theorem 2.3.1, we want t o o b t a i n a

convergent subsequgnce of i n i t i a l p o i n t s

t

E (b

- &,b+ & ) ,

monotone i n

or

$A(b)

$A(b)

5

-< 0

t

E

(b

i s monotone i n n since {Pn] i s n and BVP's a r e unique. Also, e i t h e r $;(b) $ (t)

n

f o r i n f i n i t e l y many n ' s .

[a similar argument i s v a l i d i f

monotone decreasing i n

b

-< sn 5 b +A(b)

+ s

<

Let

E

n

+so,

monotonicity of Jln(sn), +A(",)

t*

E

$

n

[b,b

E

Let

+ E).

> -

0

Assume

qn(t)

is

Then

From t h e mean value

sn = min S

n'

Then

K < $;(sn) < 0. Thus, without l o s s o f $A(",) +qo. If $;(b) = 0, t h e n b = s

and

0, q n ( t )

However, then

t*

_> $A(t) 2 K].

i s not empty.

Sn

generality If

n].

>b, 0 Sn = [ t : t -

theorem,

We may assume

$'(b) > 0. n i s monotone increasing i n n f o r

qn(t)

- &,b + & )

Define

For f i x e d

A similar argument holds when

0.

also that

(xn,yn).

<

Sn

q

until and thus

we have

$'(t*)= 0 n qn(sn)

JIl(sn)

< q.

< JI,(sn).

a r e bounded condition (2.4.4)

t* > b.

f o r some From t h e Since implies

l i e i n a bounded region.

{xn(sn),yn(sn)] Choose a convergent subsequence

115

sn + so, x (s ) n n

--f

x0'

n

.

2 . SHOOTING TYPE METHODS

yn(sn) + y o . where

6

Let

be a b a l l of radius

Ss

- so.

6 about

For large n, (sn,xn,yn)

E

S6.

(so,xo,yo) Since a l l

[a,so + 61, we have ( x n ( a ) , y n ( a ) ) E Sk s u f f i c i e n t l y large. Let Pn = ( x n ( a ) , y n ( a ) ) , Pick

solutions e x i s t on for

k

Ql,%

E

S1

llqll > k, Ilgll > k,

such t h a t

rb

From the montonicity of rb(Q2)

> rb(Pn)

f o r a l l n.

either

rb(%)

Thus e i t h e r

Q1 > Pn > %. > rb(Pn) o r rb(%) 2 q o r and

rb(&2) 2 q. This i s a contradiction t o t h e maximality of and the proof follows. COROLLARY 2.4.1.

q

Consider t h e BVP

(2.4.5)

X"

,

= h ( t X, x ' ),

(2.4.6)

r x(a) 1

+

r2x1(a) = c 1'

(2.4.7)

s x(b) 1

+

s 2 x ' ( b ) = c2,

2

where h E C[[a,b+E)xR ,R]

and assume t h e BVP (2.4.5),

(2.4.6) and s 1x ( t ) + s 2 x ' ( t ) = c has a t most one solution on [ a , t ] f o r each t E ( b - E , b + & ) and a l l c E R. I f

[s x + s 2 x 1 I 2 + [s x ' + s 2 h ( t , x , x ' ) ] 2 1 1 uniformly f o r

t

E

solution of (2.4.5) 2.5

[b

- E , b + E],

- (2.4.7).

+m

as

x2

+ xf2 +m

then there e x i s t s exactly one

ANGULAR FUNCTION TECHNIQUE

I n the preceding sections of t h i s chapter, we used a shooting type method t o prove t h e existence of solutions from uniqueness assumptions.

I n what follows, we wish t o employ

the angular f'unction technique t o study a number of r e s u l t s concerning existence and uniqueness i n a unified way.

In

p a r t i c u l a r , we are i n t e r e s t e d i n discussing the solutions of the d i f f e r e n t i a l equations

116

2.6. FUNDAMENTAL LEMMAS

(2.5.1)

Y' = g ( t , x , y )

x ' = f(t,X,Y),

s a t i s e i n g the boundary conditions (2.5.2)

.(a)

(2.5.3)

x(b) s i n p - y ( b ) cos f3 = 0.

Here we assume

s i n a - y ( a ) cos CY = 0 ,

2

f,g E C[[a,b] xR ,R],

0

-< CY < T

and

0

< f3 5

T.

We may geometrically i n t e r p r e t t h e problem as finding a solution z ( t ) = ( x ( t ) , y ( t ) ) of (2.5.1) which l i e s on the s t r a i g h t l i n e x s i n . a , y cos CY = 0 a t t = a, and on the l i n e

-

x s i n B - y cos B = O

at

t=b.

With respect t o t h e solution polar angle cp(t)

z ( t ) = ( x ( t ) , y ( t ) ) the

i s defined i n the

q(t)

xy plane.

i s called the angular function of

defined as long as formulas r e l a t e

cp(t)

and

z(t)

x ( t ) = Ilz(t)ll cos d t ) ,

=

and i s well

z(t)

~ ( t )does not vanish.

The function

Thus the following

(x(t),y(t)):

Y(t) = llz(t)II sin cp(t),

and

cp'(t>= Observe a l s o t h a t (2.5.3)

x(t)y' ( t )

- y(t)x' ( t )

x 2 ( t ) +Y2(t) z(t)

i s a solution of the BVP (2.5.1)-

i f and only i f i t s angular function

cp(a) = a, f o r some integer

q(t)

satisfies

cp(b) = B + k-rr

k.

The following lemma i s a version of Lemma 2.1.1.

117

2. SHOOTING TYPE METHODS

LEbW 2.6.1.

[a,b].

s

Let

Assume a l l s o l u t i o n s of (2.5.1) e x i s t on 2 be m y compact connected s e t i n [a,b] X R

.

Then t h e s e t o f a l l s o l u t i o n s passing through 2 connected s e t i n R

.

form a compact,

S

Consider now t h e i n i t i a l conditions (2.6.1)

x(to) =

y ( t o ) = YoJ

xoJ

and denote t h e s o l u t i o n s of (2.5.1) and (2.6.1) a s

z(t,z0) =

( x ( t J x 0 J ~ o ) JY ( ~ J X ~ J Y ~where ))J 0 = (xoJy0)* ~ e t~ ( t , z o ) denote t h e angular function of any s o l u t i o n of (2.5.1) s a t i s % ing (2.6.1).

Observe t h a t

cp(t,zo),

f o r each

t

E

[aJbIJ is

multivalued s i n c e s o l u t i o n s a r e not i n g e n e r a l wniquely defined We now present a lemma s i m i l a r t o

by t h e i n i t i a l conditions.

2.6.1 f o r angular functions.

-ma

LEMMA 2.6.2.

Let

be any compact connected s e t o f

S

R2

such t h a t t h e r e e x i s t s a l i n e through t h e o r i g i n which does not intersect all

t

E

S.

Assume t h a t f o r a l l

[a,b].

#

0

for

Then 2

u

0

forms a segment f o r each EXERCISE 2.6.1.

Hint: -

zo E S, z ( t , z o )

dtJzo)

ES

t

E

[a,b].

Prove Lemma 2.6.2.

Assume t h a t t h e r e s u l t i s not t r u e and use Lemma

2.6.1 t o o b t a i n a contradiction. LEMMA 2.6.3.

There e x i s t functions

m(u)

and

that m(IIzoII) _< IIZ(tJZO)ll _< ~ ( l l z ~ l l ) ~

118

M(u)

such

2.6. FUNDAMENTAL LEMMAS

where

lim m(u) =

u+ Proof:

+m,

+m

Define

and m(u) =

inf IIz(t,zO)Il. llzo II 2 u

t c [a,b I By Leinma 2.6.1,

M(u)

lhu++w m(U) =

show

and +m.

a r e w e l l defined.

m(u) Let

ro

>0

be given.

t h e s o l u t i o n s of (2.5.1) passing through t h e s e t

We now Consider

S =[(t,z):

E [ a , b l , I/zII 5 ro]. By Lemma 2.6.1 t h e r e e x i s t s a u > 0 0 such t h a t llz(t)l/ 5 uo where ~ ( t )passes through S. Hence

t

if

I/zoII > uo,

then

> r0

for

thus

m(u)

-

llz(t,ZO)ll > ro f o r a l l t E [a,b] u

> u 0'

and

This proves t h e lemma.

We now s t a t e t h e following known r e s u l t on t h e semicontinuity behavior of s o l u t i o n funnels. IEMMA 2.6.4.

Assume t h a t s o l u t i o n s of

r' where

F

E

n n C[[a,b] X R ,R

there e x i st s

6

>

0

6(t)

e x i s t on

= F(t,r)

i s continuous and

solution r 2 ( t )

of

[a,b].

For each

such t h a t f o r each s o l u t i o n rl

where

1,

= F(t,r),

r ' = F(t,r),

satisfying 119

+

rl(t)

6(t),

116(t)II < 6, where

there i s a

r ( a ) = r,(a), 1

E

of

>0

2. SHOOTING TYPE METHODS

- r 2 ( t ) II < E

Ilr,(t) f o r a l l t E [a,b]. We s h a l l now compare (2.6.2)

x' =

(2.5.1) with

N

Y ' = &,X,Y),

f(t,X,Y),

2 f , g E C[[a,b]xR ,R].

N

where

N

side of (2.6.2)

Assume also t h a t t h e right-hand

are p o s i t i v e l y homogeneous, t h a t is, N

N

(2.6.3)

f ( t , c x , c y ) = c?(t,x,y),

for a l l

c

2 0.

We write

xg(t,X,Y)

[f,g]

@;(t,cx,cy)= cg(t,x,Y)

2 {F,;i?

- Yf(t,X,Y) 2 &(t,X,Y)

if N

-Yf(t,x,Y).

We now present a comparison r e s u l t . LEMMI1 2.6.5.

L e t solutions

(2.5.1) and (2.6.2), never vanish.

Let

z(t)

and 2 ( t )

of systems

respectively, be defined on cp(t)

and

2 t )

[a,b]

and

be the respective angular

f'unctions of t h e solutions and assume

v(a> 2 5(4,

(2.6.4)

Cf,d ,> C G I .

Then

V ( t > 2 6m(t) where (2.6.5)

for

a l l t E [a,b],

N

(pm(t) i s the minimal solution of $ ' ( t ) = z ( t , c o s $,sin

$)

120

N

N

N

cos cp-f(t,cos cp,sin

5) s i n $.

2.7. EXISTENCE

The theory of d i f f e r e n t i a l i n e q u a l i t i e s then gives

The proof i s complete. If solutions of (2.5.1)

and (2.6.2)

a r e uniquely determined

by i n i t i a l conditions, then we may conclude t h a t Observe a l s o t h a t

?(t,O,O)

i s a solution of (2.6.2).

g(t,O,O)

E

E

0

cp(t) 1 ;(t); so t h a t z ( t ) E 0

Moreover, we notice, from the

homogeneity t h a t Il?(t,x,y)II 5 A (x2 + y2)*, llz(t,x,y)/l 5 I 2 A (x + y2)2 f o r some A > 0. This implies t h a t a l l solutions of (2.6.2) e x i s t on

[a,b].

From the calculations used i n t h e

previous l m a , we see t h a t the angular function s a t i s f i e s a d i f f e r e n t i a l equation independent of t h e solution, so it i s consistent t o discuss the angular function of a system with a given i n i t i a l angle. 2.7

EXISTENCE Before proving existence r e s u l t s f o r (2.5.1)

- (2.5.3),

it

i s necessary t h a t c e r t a i n r e s t r i c t i o n s be placed on the solutions of t h e comparison equation (2.6.2) which s a t i s f y t h e boundary conditions (2.5.2)

and (2.5.3).

When solutions of (2.5.1)

and

(2.6.2) are uniquely determined by i n i t i a l conditions, these r e s t r i c t i o n s reduce t o t h e following:

l e t ?+(t)

denote the angular functions of (2.6.2) conditions

-4-

cp (a) = a, ?-(a) = CY

121

+

T

and ? ( t )

s a t i s f y i n g the i n i t i a l

and suppose t h a t

2 . SHOOTING TYPE METHODS

f3

+ kT<

y ( b ) < f3

+ (k+l)T

and

f3

+

( k + l ) T < $-(b) < p

f o r a certain integer

k.

+ (k+2)~

This condition, i n f a c t , implies

t h a t t h e only s o l u t i o n o f t h e BVP (2.6.2), Z(t)

=

(2.5.2),

(2.5.3) i s

0.

Since we do not assume t h e uniqueness of s o l u t i o n s , t h e previous conditions w i l l be i n terms of minimal and maximal solutions. Consider t h e a u x i l i a r y systems

(2.7 1)

x'

= :(t,x,Y),

Y ' = b(t,X,Y)

x'

= C(t,X,Y),

Y ' = d(t,X,Y),

and

(2.7.2) where in

a,b,c,d

x and y.

2 C[[a,b] X R ,R]

E

Let

8(t)

of (2.7.1) and (2.7.2).

and

and a r e p o s i t i v e l y homogeneous $(t)

be t h e angular flmctions

Then a computation similar t o one i n

Lemma 2.6.5 y i e l d s

(2.7.3)

€ I t ( t ) = b ( t , c o s @ , s i n 8 ) cos e - a ( t , c o s B,sin 8) s i n 0,

and

(2.7.4)

$ ' ( t ) = d ( t , c o s $ , s i n 9) cos $ - c ( t , c o s $ , s i n $) s i n $.

I n order t o avoid r e p e t i t i o n , l e t us l i s t below c e r t a i n hypotheses t h a t w i l l be needed subsequently, (H ) The m a x i m a l s o l u t i o n s $ i ( t ) and $ i ( t ) 1 . satisfying and +-(a) = CY + T obey M (a) = M

$i(b) p

f o r t h e same i n t e g e r

k

@:(t)

+

=

IT obey

0 i ( b ) > f3

and

+

(k+l)r

a s i n (Hl);

(3)The following i n e q u a l i t y holds between

(2.5.1),

(2.7.1),

b(tJxJy)x

- a(tJx,y)y +i(tJxJy)

t

1. g(tJxJy)x-

-

E

(t,x>y)y

- c ( ~ J x J Y ) +Y% ( ~ J X J Y ) ,

[a,bl.

Observe t h a t when solutions a r e unique (H1) t h a t t h e only s o l u t i o n of (2.7.l),

(2.5.2),

systems

and (2.7.2):

< d(t,x,Y)x

uniformly f o r

of (2.7.3)

and e i ( t )

(2.5.2),

and

(2.5.3)

( 5 )imply

and (2.7.2),

(2.5.3) i s t h e t r i v i a l s o l u t i o n .

We now s t a t e and prove our main existence r e s u l t . THEOREM 2.7.1.

(2.5.1)

- (2.5.3)

Proof:

Under t h e hypotheses (H1)

-

(3), t h e BVP

admits a solution.

Let

z(t,c) = (x(t,c),y(t,c))

be any solution of

(2.5.1) sat i s fying t h e i n i t i a l condition (2.7.5) Let

cp(t,c)

x ( a ) = c cos a,

y ( a ) = c s i n a.

be the angular function of cp(a,c) =

("

IT

123

z(t,c)

if

c 7 0,

if

c
such t h a t

2. SHOOTING TYPE METHODS

From (€$), we o b t a i n 'pf(t,c)

(2.7.6)

5 b(t,cos

(&sin

cos v - a ( t , c o s

'p)

'p,sin

s i n cp

'p)

+ B0(t),

2 2 _s(t,x(t,c),Y(t,c))/(x ( t f c ) + Y ( t j c ) ) . be t h e minimal s o l u t i o n s of

where 60(t) -+ e m ( t ) , g-(t) m

rv

;l(t)

(2.7.7)

Let

g)

= b ( t , c o s 0 , s i n 0 ) cos z - a ( t , c o s 5 , s i n

s i n ;i'

+ 60(t),

-+ 0 (a)

where

=

m

a, g-(a) m

= cf

+ T, it follows t h a t

From (2.7.6) and t h e proof o f Lemma 2.6.5,

(%),

In view of

>

0:(b)

(2.7.9)

f3

we may p i c k

+ krr +

@-(a) =

cf

(2.7.10) whenever

Oi(b) > B + ( k + l ) T +

E,

6>0

so t h a t t h e r e

e+(t), 0-(t)

o f (2.7.3),

where

0 (a) = a,

p

>0

E

satisfying

- e-(t>I

le;(t)

and

< 6 for a l l t

I6,(t)I

+

that

- e+(t>l < E

Iz:(t)

[a,b]. Bo(t)

<

IcI

llz(a,c)I(

> ro,

We thus obtain, using (2.7.8) (2.7.11)

'p(b,c)

> 0:(b)

then

6 if

124

b(t,c)

- (2.7.10),

-> e+(b) - E > 0:(b)

<

E,

7I

Because o f (

MOreover, Lemma 2.6.3 implies t h a t t h e r e exists an such t h a t , i f

E.

we may p i c k

+ TT,such

there exists a

> 0 so small t h a t

(%),

By Lemma 2.6.4 and exist solutions

E

llz(t,c)

> P.

ro > 0

11 > P.

t h a t for

-E

),

>B+

c

krr,

2 ro

2.7. EXISTENCE

By applying a s i m i l a r analysis t o t h e maximal solutions of

(2.7.4) and using (H1)’ we i n f e r the existence of that, for

c

2 rl

(2 7.13

q(b,C)

r1

>0

)

l i e on

such

< B + (k+l)Tf~

and f o r c

(2.7.14)

5 -r 1 (k 4-2)T.

q(bJC) < @

Hence, f o r

c

2 r3

max(r J r ), 0

1

B + ki-r < cp(b,c) <: B + (k + l ) ~ , and f o r

5 -r3

c

B + ( k + l ) n < q(b,c) < This a s s e r t s t h a t the points

@

+ (k+2)7r.

z(b,r )

3

d i f f e r e n t sides of the s t r a i g h t l i n e Consider the s e t of solutions

2.6.1, is

x sin

3 3

such t h a t

@ - ycos @ = 0.

Thus

z(b,-r

x sin @

-y

uIlz I1 <: r3 [ z ( t , c ) 1.

t h i s s e t i s connected i n

c* E (-r , r )

and

R2-for each

z(b,c*) z(t,c*)

3

cos f3 = 0.

t.

BY

Lemma

Hence there

l i e s on the s t r a i g h t l i n e i s a solution of the BW.

This concludes the proof of Theorem 2.7.1.

We leave the proof of t h e following theorem as an exercise since i t s proof i s very similar t o t h e proof of Theorem 2.7.1. 13IEOREM 2.7.2.

(H2) holds. and

ei(t)

Consider the system (2.7.1) and assume

Moreover suppose t h e maximal solutions of (2.7.3)

satisfying

125

+ OM(&)=

cy

and

ei(t) @;(a) = a +7r

2 . SHOOTING TYPE METHODS

e+(b) < f3 + ( k + l ) n and M e x i s t s a s o l u t i o n of

obey

EXERCISE 2.7.1. COROLLARY 2.7.1. (Theorem 2.7.2)

ei(b)

< f3

f

(k+2)T.

Then t h e r e

Prove Theorem 2.7.2. Under t h e hypotheses of Theorem 2.7.1

t h e r e e x i s t s s o l u t i o n s of (2.5.1)

[(2.7.15)]

s a t i s f y i n g t h e boundary conditions x(a) s i n a

(2.7.16)

- y(a)

cos CY = c

x ( b ) s i n p - y ( b ) cos

f o r any

c1,c2

E

Hint: -

1' c2

R.

EXERCISE 2.7.2.

bounded by

f3 =

Prove Corollary 2.7.1.

U s e t h e f a c t t h a t a homogeneous expression i s 2 A(x + y2)' and make a change of v a r i a b l e s by

t r a n s l a t i n g t h e p o i n t of i n t e r s e c t i o n of t h e two l i n e s i n (2.7.16) t o t h e o r i g i n .

Then apply Theorem 2.7.1 (Theorem

2.7.2). EXAMPLE 2.7.1.

The equation x"(t)

+a

sin x + h ( t )

= 0

has a s o l u t i o n which s a t i s f i e s t h e condition

x(a)

= x(b) = 0 .

To show t h i s consider t h e s o l u t i o n s of XI'

= 0,

x ( a ) = x ( b ) = 0.

It i s clear t h a t t h e only s o l u t i o n of t h i s BVP i s t h e t r i v i a l

126

2.8. UNIQUENESS

solution x ( t )

as

1x1 + I x ’ (

=

0.

bbreover

(a s i n x + h ( t ) ( / ( ( x l+

(XI

I)

+O

An immediate application of Theorem 2.7.2

+m.

y i e l d s the desired r e s u l t . EXERCISE 2.7.3.

Show t h a t t h e boundary value problem

x” + g ( t , x , x ’ ) = 0,

x ( a ) = x(b) = 0,

has a solution, i f Ig(t,x,x’)1 2.8

1. Alxl +

B,

where

E

C[[a,b] xR+,R]

A(b-a)

2

<

2

IT

.

UNIQUUENESS

Assume t h a t

Li,

Mi,

Ki,

Ni

homogeneous i n t h e i r second variable. conclude t h a t

127

and a r e positively

From (2.8.1), we may

2. SHOOTING TYPE METHODS

Define now the Functions

128

2.8. UNIQUENESS

uv

-> 0,

uv

<

uv

> 0, -

uv

<

UV?

uv

<

UV?

0,

0, 0, 0, 0,

uv < 0. Consider t h e comparison systems (2.8.2)

u'

= P1(t,u,v),

v' = P2(t,u,v),

(2.8.3)

U'

= Ql(t,U,v),

V'

Observe t h a t u

and

v.

= &(t,u,v).

Pi, Qi, i = 1 , 2 a r e p o s i t i v e l y homogeneous i n + Letting $ ( t ) , $ - ( t ) be t h e angular f'unctions

associated with (2.8.2) and

+

8 (t), e-(t)

be t h e angular

functions associated with (2.8.3), we have (2.8.4)

$ ' ( t ) = Pl(t,cos Fl(t,

(2.8.5)

$1,

e i ( t ) = Ql(t,cos @ , s i n e ) cos e - & ( t , c o s @ , s i n e ) s i n 3

Since

$,sin +) cos + - P 2 ( t , c o s $ , s i n JI) s i n JI

Pi,

Q.

1

F2(t,8).

a r e continuous everywhere except along t h e co-

o r d i n a t e axes, we agree t h a t s o l u t i o n s of (2.8.2), d i f f e r e n t i a b l e almost everywhere.

(2.8.3) are

A s i m i l a r statement holds f o r

t h e angular functions which a r e not d i f f e r e n t i a b l e along t h e angular rays

n~r/2, n = 0 , +I, 2 2 ,

... .

We a r e now a b l e t o s t a t e and prove a uniqueness r e s u l t .

129

e

2 . SHOOTING TYPE METHODS

!EBOREN 2.8.1.

Assume

t h a t conditions (H1) and

(2.8.9, respectively.

f

and g

(5)hold

s a t i s f y (2.8.1) and

r e l a t i v e t o (2.8.4) and

Then i f solutions of (2.5.1) a r e

uniquely defined by the i n i t i a l conditions, there e x i s t s one solution of t h e BVP (2.5.1)- (2.5.3).

then z ( t ) = (;(t),y(t)) (2.5.2),

(2.5.3).

satisf'y the boundary conditions

Also l e t

angular function of -~ ( t ) . For convenience, we s h$atl)l l ebet the G(t), = y(t), -z = T ( t ) , and = $ t ) . We claim t h a t -x%(t,z,F) -T€$(t,x,y) 5 FIG-GIT (2.8.6) -< 5;P2(t,<,7) - y P p , Z , T ) . 3

From our previous estimates

-x(%(t,;) -

-

x ( % ( t , x ) +N2(t,y)) if i;

= xP2(t,Z,T). Similarly,

130

+N2(t,y)) if 2

y 2 0, ? < 0,

2.8. UNIQUENESS

The lower estimates i n (2.8.6) a r e obtained i n t h e same manner. Hence we obtain from (2.8.6), (2.8.7) where

F2(t,p)

5 7' =

6 y1-: z')/(s;2+y2)5 Fl(t,$),

F1, F2 correspond t o (2.8.4) and (2.8.5), respectively.

#

Notice t h a t ( z ( a ) , y ( a ) )

(0,O)

by t h e uniqueness of

i n i t i a l value problems and t h a t t h e r e e x i s t s no y ( t o ) = ( x ( t o ) , y ( t o ) ) = (0,O).

such t h a t

to

E

[a,b]

This implies

-

cp(t)

i s w e l l defined. From (H1),

(H2), and (2.8.6) and the theory of d i f f e r e n t i a l

inequalities, which i s applicable since continuous everywhere except a t

Fl(t,%),F2(t,5)

0 = nn/2, n = 0, 51,2 2 ,

are

...,

we obtain, using exactly t h e same methods as i n the proof of that either

"heorem 2.7.1, f3

+

krr < F(b) < p

+

(k+ l)a

if

cp(a) =

CY

+

( k + 2)n

if

q(a) =

CY

or

p

+

( k + 1)n < p ( b )

I n e i t h e r case

<

f3

x(b),y(b)

+ n.

cannot s a t i s f y

-

x(b) s i n p - y ( b ) cos p = 0 ,

unless ( z ( b ) , y ( b ) ) E 0,

a contradiction.

The existence of a

solution follows from t h e next corollary.

COROLLARY 2.8.1.

Assume t h a t t h e hypotheses of Theorem

2.8.1hold with t h e exception of the uniqueness of solutions Then there e x i s t s a solution of

of i n i t i a l value problems. the BVP (2.5.1) Proof:

(3) holds.

- (2.5.3).

I n view of Theorem 2.7.1, Let

x2 = y2

E

0

and

131

it s u f f i c e s t o show

x 1 = x, y1 = y.

Then we

2 . SHOOTING TYPE METHODS

o b t a i n from (2.8.6)

Letting

we see t h a t

G(t,x,y) = xg(t,O,O) -yf(t,O,O), G(t,x,y)/(x

lim

2 2 x +y + m

t

uniformly f o r

E

[a,b].

2

+y

2

1=

0

Thus (H ) holds and t h e proof of

3

Corollary 2 . 8 . 1 i s done.

A s before, our r e s u l t s immediately apply t o t h e more

g e n e r a l boundary conditions (2.7.16).

THEOREM 2.8.2. (Corollary 2.8.1)

Under t h e conditions of Theorem 2.8.1 t h e BVP (2.5.1),

(2.7.16)

has e x a c t l y one

s o l u t i o n ( a t l e a s t one s o l u t i o n ) . EXERCISE 2.8.1.

Prove Theorem 2.8.2.

We now apply our r e s u l t s t o t h e case i n which 2

C(l)[[a,b] xR ,R].

f,g E

I n o r d e r t o do so we must consider f i r s t

t h e case of l i n e a r systems; namely, t h e system x ' = all(t)x

(2.8.8) where

a

ij

E

+ a12(t)y,

C[[a,b],R].

(2.8.9)

e l ( t ) = a21 cos

8 = e(t)

Let

a s s o c i a t e d with (2.8.8).

Y ' = agl(t)x

a22(t)~,

be t h e p o l a r angle

Then 2

F3(t,e).

8

+ (a22 - a 11)

cos 0 s i n 8

Assume now t h e r e e x i s t two l i n e a r systems (2.8.10 )

+

z' = A(t)z,

z ' = A(t)z,

132

- a 12 s i n 2 8

2.8. UNIQUENESS

where

ma

Letting

Cp,

z t = x(t)z

JI be the angular functions associated with and

z7 = A(t)z,

V(t)

F3(t,cp)

respectively, we obtain, as i n

(2.8.91, (2.8.11) Since and

f

and g

and

J17(t) = F3(t,q).

a r e d i f f e r e n t i a b l e we have, f o r

(x2,Y2),

where

133

(xl,yl)

2 . SHOOTING TYPE METHODS

f o r a l l t E [a,bl and -m < x,y,u,v uniqueness statement i s true.

<

Then t h e following

+m.

COROLLARY 2.8.2. Assume i n (2.5.1) t h a t f , g E 2 C(l)[[a,b]xR ,R] and t h a t there e x i s t systems (2.8.10)

satisfying (2.8.14).

Assume a l s o t h a t t h e solutions

cp(t),

of (2.8.11) s a t i s f y (H1) and (H2), respectively.

Jr(t)

the B W (2.5.1), Proof:

(2.7.16) has exactly one solution.

Replacing

x

by

x2

+

s(yl-y2),

and integrating (2.8.14)

respect t o

s,

Hyptheses (H1),

Then

s(x1-x2)' from

0

y to

by

y2

1 with

we obtain

(%),

and (2.8.15)

imply t h a t t h e r e e x i s t s no In

n o n t r i v i a l solution of the BVP (2.8.8) and (2.7.16). (2.8.12) and (2.8.13) then (2.5.1)

where

+

let

x2 = y2 = 0

and

x1 = x, y1 = y;

reduces t o

6 ( t ) = f(t,O,O)

3

s3(t)/(x2+y2)

-10,

and

s 4 ( t ) = g(t,O,O).

s4(t)/(x2+y2)

134

-10

as

Hence x2

+

y2

-+ma

2.8. UNIQUENESS

Because of Theorem 2.7.1, (2.7.16)

t h i s assures us t h a t (2.8.16),

o r t h e equivalent problem (2.5.1),

(2.7.16)

has a

solution. Assume now t h a t t h e r e a r e two s o l u t i o n s (X1W,Yl(t)),

Z,(t)

z (t) =

1

= ( X 2 ( t 1 , Y 2 ( t ) ) of (2*5*1), ( 2 * 7 * 1 6 ) *

m e n x ( t ) = x,(t) - x 2 ( t ) , y ( t ) = y,(t) - y 2 ( t ) i s a s o l u t i o n As j u s t mentioned, t h e r e e x i s t s of (2.8.8), (2.5.2), (2.5.3). no n o n t r i v i a l s o l u t i o n of (2.8.8),

-

x(t) = y(t) complete.

(2.5.2),

(2.5.3).

Hence

"hus s o l u t i o n s a r e unique and t h e proof i s

0.

EXAMPLE 2.8.1.

We show t h a t t h e equation

x"(t)

+

6 sin x

+ h(t)

= 0

has a unique s o l u t i o n s a t i s f y i n g t h e boundary conditions .(a)

= x(b) = 0

if

I n Example 2.7.1 we have shown t h e r e e x i s t s a t least one solution.

We now apply Corollary 2.8.2 t o prove t h e r e exists

a t most one s o l u t i o n .

Observe t h a t t h e system x " ( t ) k 161x = 0

s a t i s f i e s (2.8.14)

with

.(a)

= x ( b ) = 0.

of t h i s problem a r e unique, if

according t o Corollary 2.8.2,

x(a)

= x(b) = 0

.

-

t h e equation with conditions

has a unique s o l u t i o n .

EXERCISE 2.8.2. XI'

(61<

We know t h a t s o l u t i o n s 2 Hence, T2/(b a)

+

Show t h a t t h e boundary value problem x ( a ) = x ( b ) = 0,

g ( t , x ) = 0,

has a unique s o l u t i o n i f E

5

gx(t,x)

5

2 /(b - a ) 2 )

(T

135

-

E,

E

> 0.

2 . SHOOTING TYPE METHODS

2.9

ESTIMATION O F

NUMBER OF SOLUTIONS

I n some cases, it i s of i n t e r e s t t o determine i f a BW has s e v e r a l s o l u t i o n s and t o estimate t h e number of t h e s e s o l u t i o n s . Before s t a t i n g our main r e s u l t we need some preliminary developments.

Assume t h a t t h e r e e x i s t two systems

(2.9.11

x ' = r(t,x,y),

Y ' = S(t,X,Y),

(2.9.2)

x'

Y ' = W(t,X,Y),

where

Let

8

r,s,v,w and JI

= V(t,X,Y),

E

C [(a,b] xR2,R]

and p o s i t i v e l y homogeneous.

be t h e p o l a r angles a s s o c i a t e d with (2.9.1) and

(2.9.2),

respectively.

Then

(2.9.3)

0 ' = s ( t , c o s @ , s i n 8 ) cos 8 - r ( t , c o s 0 , s i n 8 ) s i n 8

and (2.9.4)

$I

= w(t,cos + , s i n

$) cos $ - v ( t , c o s $ , s i n J I ) s i n 9.

We now assume conditions similar t o (H1) t h a t i s , assume t h a t :

( 5 )of

Theorem 2.7.1;

(HI) t h e maximal s o l u t i o n s & ( t ) and $;(-ti) of (2.9.4) 1 . s a t i s f y i n g $ ( a ) = a and *-(a) = a + 7r have t h e property M

JIL(b)

(H;) satisfying

M

< p + (a+l)n

and

t h e minimal s o l u t i o n s

+ 8 (a) = m

a

and

$&b) 0:(t)

@'(a) = a m

136

< B + (a+2)n;

and

+

T

0i(t)

of (2.9.3)

have t h e property

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

where

a r e continuous an d

Ej2

,ljt

B1 (t,x, Y 1

lim

x2+ y2 4 0

x

= o

and

lim 2 x +y24 0

+ Y

,X,Y 1

62 (t

2

x + Y

2

= 0.

We a r e now able t o estimate t h e number of solutions of the BVP (2.5.1)-

(2.5.3).

Assume t h a t conditions (H1) - (?), ( H i ) THEOREM 2.9.1. (5)a r e s a t i s f i e d . Then t h e BVP (2.5.1) - (2.5.3) has a t l e a s t 2 ( k- P, I nonzero solutions, whenever k # a .

Proof:

z(t,c) = (x(t,c),y(t,c))

Let

be any solution of

(2.5.1) s a t i s f y i n g t h e i n i t i a l conditions = c cos

.(a) and l e t

y(a) = c s i n

a,

(Y

be t h e angular function o f t h i s solution such

cp(t,c)

that

Assume now Theorem 2.7.1, an

rl

>0

c

>

Using the same pmof as t h a t of

0.

we conclude from (H1)

such t h a t f o r

p

+ kn<

c

2

cp(b,c)

0

such t h a t f o r

0

there e x i s t s

rl

Similarly, we obtain from (Hi)

r

- (3) that

< c -< r0

(k+l)n.

- (3) the

existence of an

ro < rl, we can apply Lemma 2.1.1, q(b,[ro,rl]) i s connected, The previous

Since we may assume and conclude t h a t

two i n e q u a l i t i e s on

cp(b,c)

then imply the existence of 137

2. SHOOTING TYPE METHODS

p o s i t i v e numbers

+

cp(b,c.) = p J

m

where

+

c

E

3

+

[ro,rl]

such t h a t

( j = m+l,m+2

jT

,...,m + ( k - J l ) ,

= min(k,J).

S i m i l a r l y t h e existence of negative numbers

c-

3

found f o r which cp(b,cT) = @ +

J

T +

jn

(j = m + l , m + 2

Hence t h e r e e x i s t s o l u t i o n s

+

z(t,cj)

,...,m +

and

2 ( k-

implying t h a t t h e r e a r e a t l e a s t

can be

Ik-Jl).

z(t,cT)

J

of t h e BVP

s o l u t i o n s , completing

t h e proof o f Theorem 2.9.1.

We now present conditions on system (Hj) a r e s a t i s f i e d . and s a t i s f y

Assume t h a t

f(t,O,O)

2.5.1)

so t h a t ( H i ) 2 ,R]

f , g E C'')[[a,blxR

g(t,O,O)

0.

Let

and consider t h e l i n e a r system (2.9- 5) The angular f'unction (2.9.6)

p(t)

then s a t i s f i e s

P ' ( t ) = gx(t,O,O) cos

x cos

p

2

IJ. + [gy(t,O,O) -fx(t,O,O)l

sin p - f

2

Y

(t,O,O) s i n

+

p.

+

% ( t ) = pm(t) = + pi(t) p - ( t ) where & ( t ) , p i ( t ) a r e , p ( t ) and G ( t ) Let r e s p e c t i v e l y , maximal and minimal s o l u t i o n s of (2.9.6). The s o l u t i o n s of (2.9.6) a r e unique and thus

us assume t h a t

138

-

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

a.

f o r some i n t e g e r Now l e t

cp(t,c)

be t h e angular function of any s o l u t i o n

z ( t , c ) = ( x ( t , c ) J y ( t J c ) ) of (2.5.1). x(t) c x(t,c),y(t)

E

we o b t a i n

y(t,c),

T ' ( t ) = gx(tJoJo) cos

2

Consequently, l e t t i n g

cp + ~ ~ y ~ t J o J o ) ~ f x ( t J o , O ) ~

2 x cos cp s i n cp-fy(t,OJ0) s i n cp

+

6(t),

where

Since

- gy(t,oJo)y(t)

[@;(tJxJy)-gx(tJo,o)x(t)

6(t)

g

fx(tJOJO)x(t)

- fy(tJO>O)y(t)1/(x2(t) + Y 2 ( t ) ) *

is differentiable,

6(t) + O

as

x

Hence (HI) i s s a t i s f i e d , i f we choose 6(t)(x2

+

y 2 )3 and

systems (2.9.1)

-f(t,xJy)

+

y2 - 0 .

Gl(t,x,y) Our comparison f

Gl(t,xJy).

b2(t,x,y)

2

and (2.9.2) a r e p r e c i s e l y system (2.9.5) and

the conditions (HI) and (H;) follows immediately from (2.9.7). 1 We thus deduce t h e following r e s u l t . COROLLARY 2.9.1.

f J g E C ( l ) [ [ a J b ] XSP,R] and s a t i s f y

f(t,O,O)

Assume t h a t (H1) J

where

sp

= g(t,O,O) = 0 .

- (H

) hold.

c (X,Y)3 IX2

y2

Let

5

P23 ,

If t h e s o l u t i o n s of

(2.9.6) s a t i s f y (2.9.7), then t h e r e exist a t least s o l u t i o n s of t h e BVP (2.5.1) (2.5.3).

-

21k- 41

It i s i n t e r e s t i n g t o f i n d conditions such t h a t t h e 2 I k - R 1 nonzero s o l u t i o n s .

boundary value problem has e x a c t l y

Clearly, t h i s w i l l always occur when monotonic i n

c,

both f o r

a s u f f i c i e n t condition f o r

c

>0

cp(t,c)

139

cp(t,c)

and

c

<

is strictly 0.

We now present

t o be s t r i c t l y monotonic.

2 . SHOOTING TYPE METHODS

Assume system (2.5.1) has the following

LENMA 2.9.1.

properties : 2 xg(t,X,Y) -Yf(t,x,Y) 2 0 , ( t , X , Y > E [a,bI xR 2 the paths of two d i f f e r e n t solutions z,(t) and

(a) (b)

z 2 ( t ) which s a t i s f y the conditions cp (a) = cp (a) = t a n 1 2

tan where and

-1

a

cp ( t )

1 z2(t), (c)

5

cp,(t),

and

-1 a,

cp,(t> 5

cp (b) = rp2(b) = t a n 1

B

tan-'

t

for

E

-1 p,

[a,bl,

cp2(t) a r e the angular functions of

zl(t)

respectively, do not i n t e r s e c t ; ( l / r ) [ g ( t , r cos cp,r s i n cp) cos cp

F(t,cp,r)

i s s t r i c t l y increasing i n

f ( t , r cos cp,r s i n cp) s i n cp] Then f o r each solution

z(t,c)

the angular function

i s s t r i c t l y increasing function i n

r.

cp(t,c)

c.

If the inequality i n (a) i s reversed and i f

s t r i c t l y decreasing, then EXERCISE 2.9.1.

Hint:

Let

cp(t,c)

Prove Lemma 2.9.1.

cpl(t,cl)

and

f o r two solutions

zl(t,cl)

and show

> cp;(a,c2).

t

E

cpi(a,c,)

(a,a+6)

if

F(t,cp,r) i s i s s t r i c t l y decreasing i n c.

6

cp2(t,c2)

and

be angular functions

z2(t,c2).

Assume

c1

>

c2

This inequality holds f o r

i s s u f f i c i e n t l y small.

Show t h a t the

(a,b] by using (b) and (c) and the theory of d i f f e r e n t i a l i n e q u a l i t i e s . inequality holds on

We have thus proved the following r e s u l t . COROLLARY 2.9.2.

Assume t h a t the conditions of Theorem

2.9.1 and Lemma 2.9.1 a r e s a t i s f i e d . 21k-

Then there e x i s t exactly

a ( nonzero solutions of the BVP (2.5.1)

140

- (2.5.3).

2.9. ESTIMATION OF NUMBER OF SOLUTIONS

Observe t h a t Corollary 2.9.2 remains v a l i d when i s s t r i c t l y decreasing i n

EXAMPLE 2.9.1.

cp(t,c)

c.

We o b t a i n an estimate, depending on

6,

of the number of s o l u t i o n s of t h e equation

+

X"

6 sin x

=

0,

s a t i s fying x ( 0 ) = 0,

x(1) = 0 .

W e apply Corollary 2.9.2 t o o b t a i n t h e r e s u l t . 2 Then system (2.9.5) becomes 161 < 7r

.

(2.9.8)

x ' = y,

Assume

y' = -&

with t h e boundary conditions x(1) = 0 .

x ( 0 ) = 0,

Since t h e r e e x i s t no nonzero s o l u t i o n s , it follows e a s i l y t h a t

a

= -1.

I n order t o find

k,

we observe t h a t t h e r e e x i s t s no

s o l u t i o n of x" = 0,

x ( 0 ) = 0,

x ( 1 ) = 0.

It follows e a s i l y t h a t k =-1. Thus f o r 161 <

2

7r

, we s e e (even

though k = a ) from Corollary 2.9.2 t h a t t h e r e e x i s t no n o n t r i v i a l 2 2 solution; hence t h e r e e x i s t s o n l y t h e t r i v i a l s o l u t i o n . If n T < I 61 < (n + 1)2T2, n = +1,+2, , t h e n an easy c a l c u l a t i o n y i e l d s

...

= n - 1. Hence t h e r e e x i s t s We l e a v e t h e case EXERCISE 2.9.2.

x"

+

2 [ ( n - 1)+1] 2n

1 61 = n2T2

solutions.

f o r t h e following exercise.

Determine t h e number of s o l u t i o n s of x(0) = 0 = x ( l ) ,

6 sin x = 0 ,

s a t i s fying

141

2 . SHOOTING TYPE METHODS

x ( 0 ) = x(1) = 0

when 2.10

161 =

n

2 2

T ,

,... .

n = 1,2

EXISTENCE OF IJYFINITE NUMBER OF SOLUTIONS I n Section 2.7,

we discussed t h e existence of s o l u t i o n s

of t h e BVP by showing

l i e s i n two s e c t o r s located on

z(b,c)

x s i n @ - y cos @ = 0

d i f f e r e n t s i d e s of t h e s t r a i g h t l i n e when

Ic I

i s sufficiently large.

It i s geometrically c l e a r t h a t t h e boundary value problem a l s o has a s o l u t i o n i f

z(b,c)

has a s p i r a l character when

c

tends t o i n f i n i t y . THEOREM 2.10.1.

Consider t h e BVP (2.5.1)

- (2.5.3).

Let

the inequality F(t,cp,r)

=

(l/r) [ g ( t , r cos cp,r s i n cp] cos cp f ( t , r cos cp,r s i n cp) s i n cp]

-

> E hold f o r

r

2

p0.

o > O

Suppose t h a t t h e

i n t o a f i n i t e number of s e c t o r s

such t h a t

-

t

E

[a,bl,

(t,cp), qj +

where

E

.

S1,. ,,Sm;

F(t,(P,r) =

lim r+ uniformly i n

xy plane can be divided

EL

cps ~

+a

P ~ + ~( j- = E

,...,n),

1

i s an a r b i t r a r i l y small constant.

Then

t h e BVP has an i n f i n i t e number o f s o l u t i o n s . Proof:

By Lemma 2.6.3,

there exists

142

co

>

0

such t h a t

2.10. EXISTENCE OF INFINITE NUMBER OF SOLUTIONS

z(t, c) function and a

p1

lo

1)

2

E

and t h e r e f o r e t h e angular

of t h i s s o l u t i o n i s w e l l defined.

cp(t,c)

Pick any

(2.

1 c 1 > co

does not vanish f o r

>

M > 0.

and choose

0

There e x i s t s

{'pj];=l

so t h a t

p0

2

F(t,cp,r)

r 2

holds f o r a l l

I5

,pl

I C P - CP.

M

where

-< t -< b,

a

M,

E.

J Define t h e discontinuous function H(CP) =

{

for

(61 5

c p = cp. -t 6 ,

J

E,

all o t h e r values of

cp.

\

We s e e immediately F(t,cP,r)

(2.10.2)

for

t

E

[a,b]

2

r

and

> H(V)

pl.

Consider t h e s o l u t i o n

$(t)

of +(a) = a-

+ ' ( t ) = H(+), Observe t h a t t h e time assume t h e value

ty

z =

Hence

$(b)

By Lemma

lcl

2

cp(t,c)

then

p2

is

2T

- $(a) 2 2nk,

z -10;

++a,

+

( ~ E / E ~n ) +

i n t e r v a l s of length M

z which i s required f o r $ ( t ) t o

where

k

i s t h e l a r g e s t number of

z contained i n

and thus $(b)

2.6.3,

(27r-2~n).

(l/M)

4 + m

there exists

Ilz(t,c)ll

[cp(a,c) = a, c

2

2

p,

E -0

As and

E

and

--3

0

M

-++a.

> 0 such t h a t when

Hence, t h e angular f'unction

pl.

of

p,]

[a,b]. as

satisfies t h e

z(t,c)

d i f f e r e n t i a l inequality lb(t,c)ll

CP'(t,c) = F ( t , T ( t , c ) ) ,

2

By t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s

t

E

[a,b].

Hence

cp(b,c)

4 + m

143

as

c

++m.

H(cP(t,c)). cp(t,c) _> $ ( t )

for

Also, i n a s i m i l a r

2 . SHOOTING TYPE METHODS

manner, we conclude

cp(b,c)

By Lema 2.6.2,

of

c,

as

++m

c

+-a.

t h e r e e x i s t s an i n f i n i t e number of values

both p o s i t i v e and negative f o r which cp(b,c) =

B

(mod

TI-).

Hence our BVP has an i n f i n i t e number of s o l u t i o n s .

The proof

i s complete.

EXAMPI8 2.10.1. x' where

gJ h

=

-h(y),

Consider t h e BVP y ' = g(x),

a r e continuous and odd.

= x(b) = 0

.(a) Assume

xg(x)

2,

2

Yh(Y) _> EoY2 and 1bx+,g(X)/X = +ooj limy+m h(y)/y = +a. To show t h a t t h i s problem has an i n f i n i t e number of s o l u t i o n s , we merely check t h e hypotheses o f Theorem 2.10.1.

We n o t i c e

inmediately

Consider t h e four segments

SiJ i = l J 2 , 3 , k J i n t h e

xy plane

defined by

where

E

i s any small constant less than a,b]xSi,

where

M(r)

T/4.

For

we have

has t h e property t h a t

144

g(x)/x

> M(r)

for

1x1

>r

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

and h(y)/y SO

that

> M(r)

M(r)

for

lyl

r

as

-++a

> r.

Since

M(r)

can be chosen

we see t h e hypotheses of

-++a,

Theorem 2.10.1 hold. EXERCISE 2.10.1. x"

Show t h e BW

+ x ( ~ + x ' ~ ) ~+ /h' ( t )

x(a) = x(b) = 0

= 0,

has an i n f i n i t e number of s o l u t i o n s . 2.11

NONLINEAR BOUNDARY CONDITIONS

Using techniques s i m i l a r t o those used i n Section 2.7, we i n v e s t i g a t e t h e existence of s o l u t i o n s of a two-dimensional d i f f e r e n t i a l system s a t i s f y i n g nonlinear boundary conditions. Results on t h e uniqueness and number of s o l u t i o n s may a l s o be deduced under conditions s i m i l a r t o those i n t h e previous s e c t i o n s , but we s h a l l omit them here. Before s t a t i n g our existence theorem c e r t a i n preliminaries need some mentioning.

I n p a r t i c u l a r , consider system (2.5.1).

Assume t h e r e e x i s t two functions, C(')[[a,b] W

such t h a t f o r each

xR2,R]

i s a l l of

W(t,x,y),V(t,x,y)

t

E

t h e range o f

V

and

and t h e s e t

R

(2.11.1) f o r a l l r e a l numbers (2.ll.2)

as

c1

and

c2;

IV(t,X,Y)l + IW(t,X,Y)l

llxll + llyll

--)a

uniformly f o r

The d e r i v a t i v e of V'(t,x,y)

=

av/at +

V

E

[a,bl.

along s o l u t i o n s of (2.5.1)

(t,x,y) +

av/+

t

-tm

av/:X

is

(t,x,y).f(t,x,y)

(t,x,y)*g(t,x,y)-

A s i m i l a r expression holds f o r Let

x(t)

and

y(t)

W'(t,x,y). be any s o l u t i o n of (2.5.1). 145

Now

2 , SHOOTING TYPE METHODS

T(t)

consider t h e angular function defined i n t h e

where V

VW

Fi,Gi

E

W;

and

and

(2.11.4)

plane.

Gi(t,V,W)/(IV(

uniformly f o r

t

E

p(t)

!%us

C [[a,bl XR2,R] Gi(t,V,W)

and polar radius

E

are p o s i t i v e l y homogeneous i n 2 such t h a t C[[a,b] xR ,R]

+ IW() - + 0

( V ( + (WI

as

[a,b].

Define

146

-> 0,

if

V

if

V < 0,

if

W

-> 0,

if

W

< 0,

if

V

-> 0,

if

V

<

0,

-+m

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

Ql(t,V,W) =

Fl(t,V,W)

if W

-> 0,

F2(t,V,W)

if

W

< 0.

where

v_>o, w_>oJ v10,

w
v
w10,

v
t h e comparison systems a r e

(2.11.6)

A' = &l(t,A,B),

B' = &2(t,A,B),

(2.11.7)

s'

T'

= Pl(t,S,T),

= P2(tJS,T).

As before we see t h a t systems (2.11.6)

have d i s c o n t i n u i t i e s along t h e l i n e s well as

S = 0, T = 0.

A = 0

and (2.11.7)

and

B = 0

may

as

Assume t h a t a l l s o l u t i o n s of (2.ll.6)

and (2.11.7) e x i s t on [a,b] and are continuously d i f f e r e n t i a b l e everywhere except a t t h e previously mentioned l i n e s . Observe that

P1,

P2,

%, %

a r e p o s i t i v e l y homogeneous i n t h e i r last 147

2. SHOOTING TYPE METHODS

two v a r i a b l e s . 0(t)

Let

and

ing t o (2.11.6) B(t)/A(t),

+(t)

be t h e angular functions correspond-

and (2.11.7),

where

B(t)

respectively.

Since

tan e ( t ) =

are s o l u t i o n s of (2.U.6),

and A ( t )

we have @ ' ( t ) = Q2(t,cos 8 , s i n 0 ) cos 8

(2.11.8)

-

%(t,cos @,sin e ) s i n

e.

S i m i l a r l y,

+

+ ' ( t ) = P2(t,cos + , s i n +) cos

(2.11.9)

-

Pl(t,cos

+ , s i n +) s i n

+.

Consider t h e BVP (2.11.10)

x' = f(t,x,y),

y ' = g(t,x,y),

with t h e boundary conditions V(a,x(a),y(a)) s i n a - w ( a , x ( a ) , y ( a ) ) cos

(2.11.11) where

0

CY

= 0,

V(b,x(b),y(b)) s i n B - W ( b , x ( b ) , ~ ( b ) ) cos B = 0 ,

-< CY < T

and

0

We now assume e s s e n t i a l l y t h e same hypotheses (H1)-

as i n Section 2.7.1.

Since t h e n o t a t i o n i s t h e same we w i l l

not repeat t h e d e f i n i t i o n s .

Therefore, assume t h e following:

( G ~ ) The maximal s o l u t i o n s

++ ( a )

(2.11.9) such t h a t +;(b)

<

f3

f o r some i n t e g e r (G2) such t h a t

M

+

=

(k+l)n

m

+L(t)

CY, +'(a) M and

= a

+i(b)

of

and

+

Tr

satisfy

p +

(k+l)T

f o r the same integer

and

>@ +

e:(b)

kT

k as i n (G1).

We a r e now able t o s t a t e our main existence r e s u l t .

Since

the proof i s similar t o t h a t of Theorem 2.7.1 we leave it as an exercise. For any

'MEOFEM 2.11.1.

- (2.11.3)

ing (2.11.1) (2.11.10),

(2.11.11)

R~

and W(t,x,y)

satisf'y-

t h e r e e x i s t s a solution of t h e BVP provided (G1) and ( G g ) hold. Prove Theorem 2.11.1.

EXERCISE 2.11.1.

Hint:

V(t,x,y)

For any

c

let

(x(a,c),y(a,c))

be a point i n

satisfying V(a,x(a,c),y(a,c)) = c cos a, W(a,x(a,c),y(a,c)) = c s i n CY. Letting

through cp(t,c)

(x(t,c),y(t,c))

(x(a,c),y(a,c)),

be any solution of (2.11.10)

we obtain the angular flmction

where t a n q ( t , c ) = ~(t,x(t,c),~(t,c))/~(t,x(t,c),~(t,c))

Proceed now as i n the proof of !theorem 2.7.1 and obtain f o r s u f f i c i e n t l y large

N

the two points

and

l i e on d i f f e r e n t s i d e s of t h e s t r a i g h t l i n e V s i n p - W cos B = 0

149

2 . SHOOTING TYPE METHODS

VW

i n the

plane.

Show then t h a t t h e r e e x i s t s a s e t V(a,S) = [-N,N].

S

C

R2

such t h a t

Thus t h e s e t

i s connected i n

R

2

.

The r e s u l t then follows.

L e t t h e hypotheses of Theorem 2.11.1

COROLLARY 2.11.1.

Then t h e r e e x i s t s a s o l u t i o n of (2.11.10) s a t i s f y i n g the

hold.

boundary conditions RIV(a,x($),y(a))

+ R2W(a,x(a),y(a)) = clY

R3V(b,x(b),y(b)) + R&W(b,x(b),y(b)) = c2, f o r any r e a l numbers

Ri,

i = 1,2,3,4

and

c., i 1

= 1,2.

Under t h e conditions of Theorem 2.11.1 and Corollary 2.11.1,

we n o t i c e t h a t when s o l u t i o n s of systems (2.11.8)

(2.11.9)

and

a r e uniquely determined by i n i t i a l conditions, then

and (G2) t o g e t h e r a r e equivalent t o t h e fact t h a t t h e r e

(G1)

e x i s t no s o l u t i o n s of (2.11.6)

and (2.11.7)

satisfying the

boundary conditions (2.11.11).

EXAMPLF 2.11.1. x’

(2.11.12)

=

2

x where

f,g

lf(Y)l

E

Consider t h e boundary value problem

x

(8)

+

f(y),

-Y(a) =

y ’ = 2y C1’

+o

Define

x,

w= 150

g(x);

x ( b ) = C2’

C[

v=

+

2

x -y.

2.1 1 . NONLINEAR BOUNDARY CONDITIONS

We a r e thus i n t e r e s t e d i n finding s o l u t i o n s of (2.11.12) s a t i s f y ing

V(b)

w(a) = c 1'

V

Observe t h a t curves

and k2.

V = kl

and W

and W

c

=

2'

s a t i s f y (2.11.1) intersect i n

= k2

since t h e l e v e l

R2 f o r every kl

It i s not d i f f i c u l t t o v e r i f y (2.11.2). V'

=

x'

=

x

+

W' = 2 x x ' - y '

f(y) = 2 2x

=

= 2y

v+ +

Now

f(y),

2xf(y)-*-g(x)

- 2w + 2xf(y) - 2y - g ( x ) +

= -2w

- g(x).

2xf(y)

F = F = V , 6 = 6 = f ( y ) , G = G = -2w, and 1 2 1 3 1 2 S~ = 62 = 2 x f ( y ) - g ( x ) . Since 1 2 x f ( y ) - g ( x ) I / ( 1 x 2 - y ~ + I ~ I ) - ~ o as 1x2-yl + 1x1 -+a, we see t h a t (2.11.4) holds. Systems Let

(2.11.6)

and p2 (2.11.9)

and. (2.ll.7) are then t h e same, with

=%

I

-2w.

P1

=

Ql

Because t h e s o l u t i o n s of (2.11.8)

=V and

a r e uniquely determined by i n i t i a l conditions due t o

the l i n e a r i t y of

Pi

%,

and

it i s s u f f i c i e n t t o

i = 1,2,

show t h e r e e x i s t no s o l u t i o n s of

v,

V' =

W ' = -2w,

satisfying W(a) = 0,

V(b) = 0 ;

t h a t t h i s i s t r u e i s c l e a r f o r any

a

and b.

Show t h e r e e x i s t s a s o l u t i o n of t h e BVP

EXERCISE 2.11.2. XI

=

x

+

2y s i n

y ' = (y/2) y ( a ) = cl,

+

2

x,

2 s i n x;

2 x ( b ) - Y (b) = c2.

151

2. SHOOTING TYPE METHODS

EXERCISE 2.11.3. 2.8 and 2.11,

Keeping i n mind t h e r e s u l t s o f Sections

s t a t e and prove a uniqueness theorem f o r a BVP

with nonlinear boundary conditions. 2.12

NOTES AND COMMENTS Example 2.1.1 i s

Theorem 2.1.1 i s due t o Hartman [41. taken from Lasota and Opial [81. Lasota and Opial [7] and Jackson

For Corollary 2.1.1,

[el.

r e s u l t o f Lees [l] while Exercise 2.1.2 2.1.3 may be found i n Jackson [2].

see

Exercise 2.1.1 i s a

i s of Levin [l]. Exercise

Theorem 2.2.1 i s due t o

Hartman [41.

Exercise 2.2.1 i s a c l a s s i c a l r e s u l t of De La Vallge

Poussin [ I ] .

meorem 2.2.2

i s due t o Tutaj [ l l .

i s taken from Keller [l] and Exercise 2.2.4

Corollary 2.2.2

i s based on Heidel

[l]. Section 2.3 i s adapted from Schrader and Waltman [4], while Section 2.4 is t h e work of Waltman [3]. For r e l a t e d discussion on uniqueness questions, see Bailey e t a l . [3], Shampine [ l ] and Sherman [ l ] . The contents o f Sections 2.5

- 2.10

a r e based on t h e work

of Perov [ l ] and Krasnoselskii e t a l . [2]. on existence and uniqueness

, s e e Waltman

due t o Bernfeld and Lakshmikanthan [2].

152

For r e l a t e d r e s u l t s

[3 1.

Section 2. ll i s

Chapter 3 TOPOLOGICAL METHODS

3.0

INTRODUCTION

I n t h i s chapter, various t o p o l o g i c a l p r i n c i p l e s a r e u t i l i z e d i n solving boundary value problems.

For example, t h e

Wazewski's t o p o l o g i c a l method t o g e t h e r with t h e connectedness properties of s o l u t i o n funnels i s used t o prove t h e existence of s o l u t i o n s .

I n t h i s s e t up, t h e boundary conditions consist

of s e t s which, i n p a r t i c u l a r cases, can be constructed from the knowledge o f upper and lower s o l u t i o n s and Nagumo's condition. We a l s o develop t h e Wazewski-like method f o r boundary value problems associated with contingent equations and s e t valued d i f f e r e n t i a l equations.

F i n a l l y , t h e continuous dependence of

solutions on boundary d a t a i s discussed.

3.1 SOLUTION FUNNELS We s h a l l consider t h e existence of s o l u t i o n s of boundary value problems a s s o c i a t e d with t h e two-dimensional system

153

3. TOPOLOGICAL METHODS

Let

f(t,x,y)

and g ( t , x , y )

be continuous i n a s e t

E

which

i s open r e l a t i v e t o

Q(to) where Q(to) I 7 R C E. tl,t2 2 0 and l e t S be a subset of C(tl). Denote by IE(S,tlJt2), the s e t of a l l points (x,Y) such t h a t there Let

is a solution

(x(t),y(t))

of (3.1.1) on

[tl,t2]

i n which

( x ( t l ) , Y ( t l ) ) E S, (x(t,),Y(t,)) = (%Y> and ( t , x ( t ) , Y ( t ) ) E E This s e t , as we have seen i s called the f o r a l l t E [t,,t2].

t = t 2' Before s t a t i n g our main r e s u l t s , we s h a l l need the following hypotheses w h i c h r e s t r i c t the behavior of solutions as they

solution funnel cross section a t

cross the

cp

and

surfaces.

$

We s h a l l always assume:

'

(Hi) f o r a n t i E [OJtoIJ (Xl'Y1) I ~ ( S l J 0 , t l ) sq(t1) implies t h a t there e x i s t s a solution of (3.1.1) emanating from w i t h a trajectory which i s on or above the $-surface (tlJxl,yl)

on some right neighborhood of

t

1'

(H2) f o r a l l tl E [O,tOl, (xl,Y1) E IE(Sl~O,tl) n)lt(,,.s , implies t h a t there e x i s t s a solution of ( 3 . 1 . 1 ) which emanates w i t h a trajectory which is on or below the from (tl,xlJy1) cp-surface on some right neighborhood of tl.

Our f i r s t r e s u l t describes qualitatively the behavior of solutions

of (3.1.1) assuming (H1) and (H2) hold. THEOREM 3.1.1. C(0)

Let

which intersects (i)

S1 be a compact connected s e t i n S+(O)

and

Sp(0).

Then e i t h e r

IE(SIJO>t) contains a compact connected component

C(t) which intersects both S ( t ) and S ( t ) f o r a l l Jr cp t E [OJto], or ( i i ) there i s a solution of (3.1.1) w i t h (x(O),y(O)) E S1 having a maximal right i n t e r v a l of [O,t+] C [O,tO] such t h a t in

154

3.1. SOLUTION FUNNELS

E: (p(t,y) - 1 < x < Jr(t,y) + 11. Assume t h e r e i s no number K such t h a t f o r a l l t E [O,tO], Proof:

Let

E'

(x,y) E IEl(Sl,O,t)

i s some i n t e r v a l

[(t,x,y)

implies [O,t+]

> K for t

C

(xn,yn) E IEl(Sl,O,tn).

lynl

+a.

IyI

5

K.

Then t h e r e

Ixn I + m,

1x1

>K

or

tn + t Ixnl + m o r

Then, e i t h e r +(t,y)

and

-k

and

it

(P(t,y),

Iyn I -+ m.

then

>0

K

Pick any sequence

By t h e continuity o f

case we have

and

such t h a t e i t h e r

let

follows t h a t i f

-< K

1x1

+ + [ t - &,t).

E

E

[O,tO] such t h a t f o r any

&(K) > 0

there e x i s t s an IyI

E

Hence i n e i t h e r

lynl + m.

By a standard diagonalization process, using t h e s o l u t i o n s

{ (xn,yn)},

associated with

of (3.1.1) with

(x(t),y(t)) on

[O,t]

[O,t+)

C

Moreover,

we may construct a s o l u t i o n which e x i s t s

(x(O),y(O)) E S1

and such t h a t

ly(t)l

+.

t +t

as

+m

This i s case ( i i ) .

( t , x ( t ) , y ( t ) ) E E' c E.

Hence, t o complete t h e proof we may assume t h e r e e x i s t s a

K

number implies

>

1x1

Let

such t h a t f o r a l l

0

-< K

t

T be the s e t of a l l p o i n t s the set 0 E T

[O,tO] such t h a t ,

E

IEl(Sl,O,t)

C ( t ) which i n t e r s e c t s both

i s nonempty s i n c e

contains a component

S J l ( t ) and

Sq(t).

Then

T

to. We

and i s bounded above by

i s closed and thus i f we l e t

s h a l l show T

(x,y) E I E l ( S l , O , t )

E (O,tO],

J y J5 K.

and

f o r a l l t E [O,T], in

t

s = sup T,

then

s = to implies t h e conclusion of Theorem 3.1.1. Let

be a sequence of p i n t s i n

{s.}

s with Ci

1

C

C(si)

S (si)

intersects both

l i m i t set of

a component of

{Ci}

Jr

and

and l e t

Then there e x i s t s a sequence where

(x;,yf)

E

C;

and

I E (S l

Let

S (s.).

' p 1 (a,b) E

L

E(x;,y;)I

converging t o

T

1'

0 si)

L

since

'

which

denote t h e L

i s nonempty.

converging t o

i s a subsequence of

{C;]

155

(a,b)

{Ci}.

3. TOPOLOGICAL METHODS

Let

be the s e t o f l i m i t points of

L'

compact and since for a l l If

intersects both

C;

then

i

{C;};

hence

S (s!) c p 1

and

is

L'

S (s;) Jr

L' intersects both S (s) and S (s).

cp Jr contains no component which intersects both

L'

S (s) and ST("), then since L1 is compact and intersects Jr both S (s) and S (s), L' i s the union of two nonempty cp compact s e t s

that

A

n L'

Assume {C!].

There e x i s t s a point

s; +s,

C

in

C(s)

closed.

a subsequence of

c (x;,~;)]

t o be the subsequence of

Thus

(p,q)

(p,q)

(p,q) E L 1 and

and we conclude

and

For i s u f f i c i e n t l y large the connected i n t e r s e c t s A and l e t (pi,qi) be the

C;.

IE,(Ci,si,s)

then

be any point i n N

(z',?') E C i ,

where

points of intersection. and since

(c,d)

Let

E M.

+ (c,d),

contained i n

such

= cp.

Choose { (x;,yy)]

1

set

J,

and N which are separated by an arc A

(a,b)

(?',Ti)

let

M

(pi,qi)

E

n S.

IEl(C;,s;,s)

which i s a l i m i t point of

i s a limit point of

(p,q) E A fl LI.

{C;].

{(pi,qi)] However,

This i s a contradiction

IEl(Sl,O,s) contains a component intersecting both S (s) and S (s). Thus T i s cp JI L'

and thus

By assumptions (H1)

and (H2) there e x i s t s a

6>0

such

that

and

IEl(C,s,t)

contains a component which ir,tersects both

S (t)

and S ( t ) f o r a l l t E ( s , s + 6 ) . However, t h i s JI cp contradicts the f a c t t h a t s = sup T. Thus s = to and ( i )

i s proven.

This concludes the proof of Theorem 3 . 1 . 1 .

Theorem 3.1.1 w i t h conditions t h a t r e s t r i c t the possibility of ( i i ) occurring can be used t o deduce existence theorems. have seen i n Chapter 1t h a t one such condition is Nagumo's condition.

The following condition w i l l thus be imposed:

156

We

3.1. SOLUTION FUNNELS

(H3) Given any number N(tO,n)

n

>0

and

to > 0 t h e r e e x i s t s a

such t h a t f o r any s o l u t i o n ( x ( t ) , y ( t ) )

(3.1.1) with

ly(O)[ < n and ( t , x ( t ) , y ( t ) ) E F, f o r l y ( t ) l < N(tOJn) f o r a l l t E [O,tO).

we have

of

t

[O,tO)

E

We now p r e s e n t an existence theorem. Assume conditions (H1)- (H ) hold.

THEOREM 3.1.2.

S1 be a compact connected s e t i n Sq(0)

and

C(to)

such t h a t

S (0). cp

Let

(x(t),y(t))

t h a t ( x(O),Y(O) )’ E S1’ for a l l t E [O,tOl. Proof: (x,y)

and l e t

S1,

E

s1

Since

l y ( t ) l < N(tO,n) (x(O),y(O)) E S1 IE(SIJO,tO)

which i n t e r s e c t s both

be a closed connected subset of

S2

S2 f l [(x,y): y

e x i s t s a s o l u t i o n of

C(0)

# $,

arbitrary]

of (3.1.1) on

( x(to)tY(t0))

E

then t h e r e [O,to]

n = sup (yI f o r

5 ).

N = N(tO,n) be a s i n (

f o r ally ( x ( t ) J y ( t ) )

Of

C ( t o ) which i n t e r s e c t s both S2

Then

(3.1.1) with

t E [OJtO]. By Theorem 3.1.1,

contains a compact connected component

conditions imposed on

such

S2 with ( t , x ( t ) , ~ ( t ) )E E

i s a compact s e t , l e t

and a l l

Let

3

S (t ) $ 0

and

insure t h a t

S2

S (t ).

cpo

n

C

The

IE(SIJO,tO)

in

# 9.

This concludes t h e proof of t h e theorem. Another approach t o t h i s problem i s t h e a p p l i c a t i o n of t h e Wazewski’s method.

We s h a l l introduce t h i s method here.

Consider t h e d i f f e r e n t i a l system (3.1.2) where

f

X’

E

= f(tJx),

n C[nJR 3, R

be an open set of closure of

x ( t 0 ) = xo

2

J

being any open set i n

OJ

Rn+l.

Let

n, anoJ t h e boundary, and E0 t h e

Ro

no.

DEFINITION 3.1.1.

A point

157

( t O J x O )E

fi

n ano

is said

3 . TOPOLOGICAL METHODS

t o be an egress point of

with respect t o t h e system

Ro

(3.1.2) i f , f o r every solution E

>0

such t h a t

x(t)

of (3.1.2),

to- E 5 t < to. An egress

for

( t , x ( t ) ) E Ro

there i s an

no i s called a s t r i c t egress point of no, Tio f o r to < t 5 to + E f o r a small E > 0.

point

( t O , x O ) of

if ( t , x ( t ) ) Denote t h e s e t of a l l points of egress ( s t r i c t egress) as

S

(S") DEFIXITION 3.1.2. topological space and B

into A

such t h a t

A

If T:

C

~ ( p =) p

i s s a i d t o be a r e t r a c t i o n of a r e t r a c t i o n of

B

onto

B

are any two s e t s of a

is a continuous mapping from

B +A

f o r every

B onto

A, A

p E A,

then

T

When t h e r e exists

A.

i s called a r e t r a c t of

B.

The following theorem of Wazewski i s quite useful. THEOREM 3.1.3.

Let

f E C[R,Rn), R

Assume t h a t through every point of solution of (3.1.2).

Let

Ro

be a nonempty subset of

r e t r a c t of

S,

there passes a unique

be an open subset of

Qo U S

but not a r e t r a c t of

at least one point

$+l R.

Suppose

no are s t r i c t egress points.

t h a t a l l egress points of Z

R

open i n

such t h a t 2.

Z fl S

Let

is a

Then there e x i s t s

no such t h a t the solution ( t , x ( t ) ) of (3.1.2) remains i n no on i t s maximal i n t e r v a l (tO,xO) E Z fl

of existence t o the right of

to.

W e now apply Theorem 3.1.3 t o the BVP prescribed i n Theorem 3.1.2.

Although we assume uniqueness of solutions of (3.1.1)

t h i s i s not e s s e n t i a l a s a Wazewski-like theorem f o r nonuniqueness has been developed.

Let

R = [(t,x,y): t

2 03 -S2;

a r e l a t i v e l y open subset of the h a l f space

no = [(t,x,y): 0 5 t < tl,

then

[(t,x,y): t

2

is 01. Let R

q ( t ) < x < q ( t ) , IyI < m1 where now a r e independent of y. Let Z f S From 1' hypotheses (H1) and (%), it i s not d i f f i c u l t t o see t h a t S = S", $(t,y), q ( t , y )

158

3.1. SOLUTION FUNNELS

and

consists of t h e union of t h e s e t s

S

We see from t h e properties of

i s a r e t r a c t of

S.

i s not a r e t r a c t of

and t h e s e t

that

S1n S

S

of (3.1.1) such t h a t

solution ( x ( t ) , y ( t ) )

S2

n S i s not connected S1 n S 1 Hence from Theorem 3.1.2 there i s a

Since

S1.

S

( tl,x(tl),y(tl))

E

and such t h a t ( t , x ( t ) , y ( t ) ) remains i n no on i t s S1 n C(t,) r i g h t m a x i m a l i n t e r v a l of existence. This implies ( t 2 , x ( t 2 ) J

due t o (H ).

Hence t h e r e e x i s t s a solution s a t i s f y -

y ( t 2 ) ) E S2 3 ing t h e BVP prescribed i n Theorem 3.1.2.

We now can obtain under t h e same hypotheses the existence of a solution remaining i n THEOREM

3.1.2, that

n

&(to).

3.1.4. Under the same hypotheses as i n Theorem

t h e r e e x i s t s a solution ( x ( t ) , y ( t ) ) of (3.1.1) such (x(O),Y(O)) E S1' ( x ( t , ) , y ( t O ) )

n

~ ( t ) E) R

Hint: -

=E

6

S ~ Jand

(t,x(t>,

Q(t,).

EXERCISE 3.1.1.

E~

R

Prove Theorem 3.1.4.

Choose a sequence of open s e t s

n

~ R+ ~~ ( t ,I ),

E~ = E,

and

n,,

W

Show t h e r e e x i s t solutions which l i e i n

where E

En boundary conditions by applying Theorem 3.1.2.

n Q(t,). and s a t i s f y t h e

= R

Then apply

Ascoli's theorem t o obtain t h e desired solution. We can now prove a r e s u l t f o r a semiinfinite BVP.

THEOREM 3.1.5.

Let

f , g E C [ [ O , ~ ) ~ R * , Rand ] assume 159

3. TOPOLOGICAL METHODS

-

kt

(H1) (H3) hold. intersecting both solution

S1 be a compact connected s e t i n C(0)

(x(t),y(t))

and ( t , x ( t ) , y ( t ) )

E

Sq(0).

e x i s t i n g on

R

n

Then (3.1.1) has a with

[O,m)

2

Q(to)for a l l t

(x(O),y(O)) cS1

0.

Prove Theorem 3.1.5.

EXERCISE 3.1.2.

Hint: -

and

S (0) $

Use Theorem 3.1.4 and a standard diagonalization

argument. 3.2

APPLICATION TO SECOND-ORDER E a T I O N S

In t h i s section we apply the r e s u l t s of Section 3.1 t o t h e case i n which

f(t,x,y) = y

i n (3.1.1).

That is, we consider

the equivalent second-order s c a l a r equation

(3s2O1)

XI1

= @;(t,X,X').

Recall t h a t a ftmction a ( t ) E C(2)[0,t

of (3.2.1)

if

(3.2.2)

a"(t) 2 g(t,(Y('(t),(Y'(t)),

bbreover,

(Y

- is >

1

t

i s a lower solution

E

[O,tOl.

becomes a s t r i c t lower solution if, i n (3.2.2),

replaced by

>.

Similarly a f'unction

i s an upper solution of (3.2.1) (3.2.3)

0

V(t)

5

B(t) E C(2)[0,to]

if

g(t,B(t),B'(t)),

t

E

[O,tOl-

If i n (3.2.3) 5 i s replaced by <, then B is a s t r i c t upper solution. We now present a general existence r e s u l t , Notice t h a t i n Chapter l o t h e r techniques have been used t o prove similar types of theorems. TI-IEOm 3.2.1. (i)

Assume t h a t :

there e x i s t s an upper solution

160

@ ( t )and a lower

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

solution a ( t ) E

of (3.2.1) with

-< B(t)

a(t)

be any open s e t containing the s e t

5x5

a(t)

S,

I

# gj

and such t h a t

with

I<

such t h a t

(x(to)Jx'(to)) E

n

with

2

y'

0

Let

5 t 5 to,

such t h a t ( x , x l ) E S1 fl S a ( 0 ) f o r some

B'(0)

be a closed connected subset of

S2

[ ( x , ~ ' ) : x'

x(t)

C(0)

' ( 0 ) f o r some

#9

Let

S2

e x i s t s a solution

Proof:

x' <~

S1 fl S (0) B

(y,y') E S1 fl Sg(0). C(to)

[(t,x,x'):

be a compact connected s e t i n

S1 fl S a ( 0 )

[O,tO].

(3) holds.

(ii) Let

Ix'

B(t)J

on

of (3.2.1)

arbitrary]

# 8.

such t h a t

(x(O),x'(O)) E S ~ ,

s2 with a ( t ) < x(t) < @ ( t ) On

We f i r s t prove t h e r e s u l t with

a(t)

Then there [Ojt0I. and

p(t)

assumed t o be s t r i c t lower and upper solutions, respectively.

(3) i s s a t i s f i e d the p o s s i b i l i t y of

Since

Theorem 3.1.1 i s eliminated. of

IE(Sl,O,t)

Let

Ct

C

which i n t e r s e c t s both

C(t)

Sa(t)

be a component and

P by

Define

P

I

B

< x1 -

(x,x') E S a ( t ) fl Ct such a l ( t ) and there e x i s t s ( y , y l ) E

ct n sB ( t ) t E [oJ51]* P

S (t).

[t;: there e x i s t s that

Notice

case ( i i ) i n

such t h a t

i s nonempty since

y'

0 E P.

-> B ' ( t )

Let

f o r a11

n = sup P.

It i s

s u f f i c i e n t t o show n = to. Exactly a s i n t h e proof of Theorem 3.1.1 it follows t h a t There e i t h e r e x i s t s

n

E

P.

We omit the d e t a i l s .

( x , x l ) E Cn fl Sa(n)

such t h a t

x = a ( n ) , x ' = a l ( n ) o r there e x i s t s

(y,y') E Cn fl S (n) B f o r i f not, then there e x i s t s y = p(n), y ' = @ ' ( n ) such t h a t nl E P. This i s impossible.

such t h a t

nl

>n

( y , y l ) E Cn fl SB (n) such The other case i s similar. I f

Consider then the case where that

y = p(n), y ' = p ' ( n ) .

161

3. TOPOLOGICAL METHODS

y ( t ) i s any solution of (3.2.1) w i t h y(n) = B(n), y ' ( n ) p'(n), then there e x i s t s an &l > 0 such that y ( t ) >

=

p ( t ) , y ' ( t ) > p f ( t ) on (n,n+E1). This follows since p ( t ) i s a strict upper solution on [O,tO]. Moreover, f o r any t > n, any solution y ( t ) of (3.2.1) w i t h y(T) = p(T), y ' ( z ) < PI(:) satisfies y(t) < p(t) for < t < % + S and y ( t ) > p ( t ) on 6 < t C t forsome 6>0. Eythecontinuityof g, it follows t h a t there e x i s t s an E~ > 0 and p l > 0 such t h a t p"(t) < g(t,y,y') for a l l t E [ n , n + ~ ~and l a l l ( y , y f ) such t h a t d((y,y'), ( p ( t ) , B ' ( t ) ) ) < pl, where d denotes t h e Euclidean metric. By standard arguments there e x i s t an &3 > 0 and p2 > 0 such t h a t any solution y ( t ) of (3.2.1) w i t h

z-

y(n) = YoJ y ' ( n ) = YA with d((YoJY~),(p(n),B'(n)))< p2 s a t i s f i e s d(y(t>,y'(t)),(p(t),p'(t))) < p1 f o r a l l t E [ n , n + ~ ~ ] . Moreover, B"(t) < y " ( t ) f o r a l l t E [ n , n + m i n [ ~ ~ , ~ ~ l . Finally, there e x i s t s an &4 > 0 such that f o r any solution

~i

where (Y1,Yi) €Cn ~ l ( t )of (3*2*1)w i t h y l ( n ) = ~ 1 y9i ( n ) = and a((Y,,Yi>,(B(n),B'(n))) < P1 both y,(t) _> B(t) and y i ( t ) 5 p ' ( t ) are not possible f o r any t E [ n , n + &4 1. If we l e t E = min[& , E ,& 1, then it follows from the 1 2 3'&4 preceding observations t h a t Ct n S p ( t ) # # and y' 2 p ' ( t ) f o r some (y,yl) E Ct fl S ( t ) f o r a l l t E [O,n+&]. A similar B argument leads t o the conclusion t h a t Ct n S 2 ( t ) # $ and x ' < a ' ( t ) f o r some (x,xt> E ct n SCY(t) f o r t E [o,n+El. Then nl = min[n+&, n + F ] E P and nl > n, a contradiction.

-

Thus n = to and we conclude that IE(Sl,O,tO) contains a compact connected component intersecting both SCy( t o ) and Sp(to). Hence S2 fl IE(SIJO,tO) # $ and hence there e x i s t s a solution x ( t )

of (3.2.1) w i t h (x(O),x'(O)) E S1, ( x ( t o ) , and ( t , x ( t ) , x ' ( t ) ) E E. Using the same type of x f ( t o ) E) S2, proof' as i n Theorem 3.1.4, we may conclude that a ( t ) 5 x ( t ) 5 p ( t ) f o r a l l t E [O,tal.

162

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

We now prove Theorem 3.2.1, B(t) 0

assuming t h a t

a(t)

a r e lower and upper solutions, respectively.

1, l e t

A (t) = a(t) - y

and

Y

and

For

+ y.

B ( t ) = B(t)

Y

Define

x >

g ( t , B ( t ) , x ' ) + x - B(t),

Q(t5 > x 5 B(t), x

g(t,a(t),x') + x-a(t),

< a(t).

Ix' I + 1 and l e t N be t h e number ( X J ' )fS1 associated with M i n ( ). Choose K = max(N, m a x Let

M = max

5

t Ero, t 0~ i

( l Q ' ( t ) l , I B ' ( t ) l > ) and define

I2K Ix' I 5 K

Ix' H(t,x,x') =

-

+

+ 2,

+ 1,

1< x '

(K+2 x ' ) G ( t , x , K + l ) ,

K

( K + 2 + x ' ) G ( t , x , - K - l),

-K- 2

<

X'

< K + 2, < - K - 1.

Eloreover A"(t) = ~ " ( t _>) g ( t , a ( t ) , ~ ' ( t ) > ) g ( t , a ( t ) , A;/(t)) - Y

Y

B"(t) = B"(t)

Y

Hence A

5

and

Y

g ( t , B(t), B ' ( t ) )

B

Y

X" =

(p,q)

(u,v)

E

union of

E

s n 1

S1,

S1

n

AY(t), A+(t)),

< g(t,

B(t), B+(t)) + y

= H(t,

By&), B;i(t))-

a r e s t r i c t lower and upper solutions of

(3.2.4) Let

= H(t,

H(t,x,x').

[ ( x , x ' ) : x = Q(o),

XI

2

B(0)l.

t h e l i n e segment from

(p,q)

[(y,yl): y = B(O),

t h e l i n e segment from

(u,v)

yi

to

and l e t

5 a'(0)1, Let

X

Y

be the

(p - ~ , q ) and

to

(u + y,v).

Consider now t h e BVP (3.2.4) with t h e boundary conditions (3.2.5)

(X(O),X'(O))

E

xY' 163

( x(t,>,x'

( t o >)

E

S2'

3 . TOPOLOGICAL METHODS

Let

E = [ ( t > X > X ' ) :O

Ix'I

<

Since

m1.

5 t 5 t O JQ ( t ) - 1 < X <

H(t,x,x') = 0

[ x ' I _> K

for

+

B(t)

+

1,

2,

we may

apply t h e proof of the f i r s t h a l f of t h i s theorem, r e c a l l i n g that

A (t), B ( t )

a r e s t r i c t lower and upper solutions Y Y respectively, t o conclude t h e existence of a solution Xy(t)

of t h e BVP (3.2.4),

< B ( t ) for - Y

X (t)

Y

for

2

Ix'I

K

+

2

(3.2.5) with t h e property t h a t A ( t ) < Y t E [O,tO]. Also, since H(t,x,x') = O i X r ( t ) [5 K

it follows t h a t

Y

+

2

on

By Kamke's convergence theorem a subsequence of converges t o a solution

Xo(t)

of (3.2.4) on

[O,tO].

n]i=l

{X

[O,tO]

11 such

that

( xo(l)Jxi)(l))

(xo(o)~xi)(o)) ' 1 ' Also A

a(t)

l/n

For

5

Xo(t)

(t) = a(t)

a(t)

g(t,x,x').

5

B(t),

- (l/n)

-< x -< B(t),

'2'

since

5 X l / n (t) 5

B(t) + ( l / n ) = Blln(t).

Ix'I < K + 1 we have

It follows, from

(?),

that

H(t,x,x') =

IXi)(t)[ < N

-< K.

Hence

s a t i s f i e s t h e BVP and t h i s concludes t h e proof of meorem Xo 3.2.1. It i s not necessary t o assume

t

and lower surface at which eventually hook

= 0

S1

i n t e r s e c t s t h e upper

provided t h e r e e x i s t solutions

S1 onto t h e lower and upper surfaces.

More precisely, we have t h e following r e s u l t .

THEOREM 3.2.2. a, B

Assume (H ) holds r e l a t i v e t o

3

E.

Let

B(t)

be lower and upper solutions f o r (3.2.1) with a ( t ) on [O,tO]. L e t S1 be a compact connected s e t i n

C(0)

and l e t

that

S2 fl [ ( x , x ' ) : x '

S2 be a closed connected subset of arbitrary]

#

@.

If

s1 such that

[(xl(u)Jxi(u))] 164

C(tO) such

(x,(t),xi(t))

and ( x 2 ( t ) , x h ( t ) ) a r e solutions of (3.2.1) with (x2(0)Jx6(0))

-<

( xl(0),xi(O)), s ~ ( u )# @

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

[(x,(V)Jx;(v))] n s 2 ( v ) # $ f o r some v E (OJt0IJ then t h e r e exists a s o l u t i o n x ( t ) of

f o r some

u

( o , ~ , I and

E

(3.2.1) with

< x(t) a(t) -

( x ( 0 ) J X ' ( 0 ) ) E S1j (X(t,),X' ( t o ) )

-< p ( t )

and

S2j

f o r a l l t E [OJtoI.

The proof i s similar t o t h a t o f Theorem 3.2.1

Proof: since

E

I (S , O , t )

E l Sp(t) for t

has a component i n t e r s e c t i n g

2 max[uJvl.

Sa(t)

and

I n Chapter 1, we have provided s u f f i c i e n t conditions f o r

(3) t o hold,

One such condition was Nagumo's condition.

We now o b t a i n a r e s u l t t o show how Nagumo's condition can be u t i l i z e d t o determine

S1 and S2.

lower and upper s o l u t i o n s of (3.2.1)

t

E

Let with

a(t)

and

a(t)

[ O J t O l . Define

A = max

I

Ia(0)

J

max

g(t,x,y)

J

Ia'(t)lJ

max lp'(t)l [OJt01

s a t i s f i e s Nagumo's condition lg(t>xJy)l

f o r a l l t E [,,to]'

-< B ( t )

IMO)- B(0) I

- B(to) I [OJt01

and assume

B ( t ) be

a(t)

5

(P(lYl)

-< x < B(t),

Letting

165

IyI

<

w

such t h a t

for

3. TOPOLOGICAL METHODS

we observe t h a t

THEOREM 3.2.3.

Assume

g(t,x,y)

s a t i s f i e s Nagumo's

a(t),B(t) E C(2)[0,to] with a ( t ) 5 B ( t ) . If S1 is any closed connected subset of C ( 0 ) such t h a t S1 fl S # 0, S1 n S4 # 0 and i f S2 is any closed 3 connected subset of C(to) such t h a t S2 n S # jd and 5 S2 n S6 # I ,then there e x i s t s a solution x ( t ) of (3.2.1) such t h a t (x(O),x' ( 0 ) ) E S1, ( x ( t o ) , x l ( t o ) ) f: S2 with condition with respect t o

a ( t >,< x ( t > 5 B ( t ) * Proof:

Choose

n

(o,xl,xi)

G

s1 n s3

and

(o,x2,x;)

E

Sq such t h a t both p i n t s belong t o t h e same component S1 of S1 n [(O,X,Y): 5 N l . Choose (O,B(O),x) ) E Sp(0) as follows. If x1 = p(O), l e t = xi; i f x1 < B(O), S1

c

choose

IY)

5 > N.

Let

%

be t h e l i n e segment joining 166

(0,xl,xi)

3.2. APPLICATION TO SECOND-ORDER EQUATIONS

to

(O,p(O),x'), 3-

where, i n t h e case when

p(O), L1 = I n a similar manner, choose ( O , ~ ( O ) , X & c )

[ ( O , X ~ , X ; )c I sl. by l e t t i n g

xk = xh

Scy(0)

x2

>a(O)

and take

( O , X ~ , X ~t )o observe

x1

x2

if

= a(0)

or

=

xc

<

-N

if

t o be t h e l i n e segment joining

L2

(O,a(O),xi).

Let

ST

= L1 U

zl

and

U L2

ST i s compact, connected, and i n t e r s e c t s both

[(O,a(O),x'): x' 5 a ' ( O ) ] and {(O,p(O),x'): x' 2 p'(0)). Pick (t , x , x ' ) E S2 n s5 and (tO>X6'Xi) E s2 n s6

-

0 5 5

such t h a t both points belong t o t h e same component S2 S2

n

[(tO,X,X'):

l i n e s given by and l e t

s;

= L

5

5 N(tO)]. Let [(t0,x5,y): y 2 x;l IX'I

u 8, u L6.

Consider the BVP (3.2.1) (3.2.6)

(O,X(O),X'(O)) E

L5

and

of and L6 be t h e half [(to,x6,Y): y

and

( t0,x(to),x'(t0))

,s;

E

5 xi1

*

S2'

By the construction of

ST and t h e assumptions on g(t,x,y), there e x i s t s a compact, connected subset C C C(to) of the funnel cross section F(tO,O,ST) = i n t e r s e c t s both

uses++ F(tO,O,s)

which

1

S p ( t o ) : y 2 p t ( t o ) ] and Hence, by Theorem 3.2.1 the al(t,)].

[(tO,x,y)

E

[(tO,x,y) E S2(t0): y 5 BVP (3.2.l), (3.2.6) has a solution

x ( t ) with

a(t)

-< x ( t ) -<

B b ) , I x ' ( t O ) l 5 N(tO) and ( t O , x ( t O ) , x ' ( t O ) E) S2* From Nagumo's condition Ix'(0)I 5 N and by t h e construction of Sr, (O,x(O),x'(O)) E S1. This concludes the proof of Theorem 3.2.3. EXA.MFTE 3.2.1.

A s an application of t h i s r e s u l t consider

the following BVP a r i s i n g i n the dynamics of c e r t a i n chemical reactions. (3.2.7) (3.2.8)

X" = -LIX'

x(0) = 0,

- L2k(x),

x ' ( 1 ) = -L1X(l).

167

3 . TOPOLOGICAL METHODS

The positive constants

and

L1

of t h e physical problem with on

[O,c]

such t h a t

k(x)

L2

k(x)

>0

for

depend on various parameters continuous and nonincreasing 0

<x
and

k ( c ) = 0.

Theorem 3.2.3 gives t h e existence of a solution x ( t ) of t h e BVP (3.2.7), (3.2.8) with 0 < x(t) < c since m(t) = 0 and @ ( t )= c are lower and upper solutions, respectively, and Nagumo's condition i s c l e a r l y s a t i s f i e d .

3.3 WAZEWSKI RETRACT METHOD "his section i s devoted t o an application of t h e Wazewski r e t r a c t method t o (3.2.1) with boundary conditions

- blxl(0),

(3.3.1)

L1(x) = rl,

L1(x)

= alx(0)

(3.3.2)

L2(x) = r2,

L2(x)

= a2x(to) +

b2x1(t0).

Without loss of generality, we may assume b ,b > 0. Let 1 2 m(t) and B(t) be lower and upper solutions of (3.2.1) such that (3.3.3)

~ ~ ( 50 ri, )

(3.3.4)

Li(B)

We s h a l l assume t h a t

g

2

ri,

i = 1,2, i = 1,2.

s a t i s f i e s Negumo's condition, t h a t is,

As an application of Theorem

1.4.1, we obtain t h e following

lemmas.

168

3.3. WAZEWSKI RETRACT METHOD

U M M A 3.3.1.

Assume t h a t

Then, f o r every

> 0,

c

t h e r e e x i s t s a constant

t h a t each s o l u t i o n x ( t )

t

E

I

satisfies Nagmots condition.

g

satisfying

a(t)

M(c)

-< x ( t ) 5

such

p(t)

for

[O,tO] and which s a t i s f i e s

C

Jx'(sol

5

c

for

some

s0 E I

a l s o has t h e property

Assume t h a t t h e r e e x i s t s a s t r i c t lower

LEMMA 3.3.2.

solution a ( t )

and s t r i c t upper s o l u t i o n

t o ( 3 . 2 . 1 ) J s a t i s e i n g (3.3.3), all t

E

(3.2.1)

(3.3.4)

p(t)

with

with respect

< p(t) for

a(t)

Then, t h e r e e x i s t s a s o l u t i o n x o ( t )

[O,tO].

defined on

[O,h+&] C [O,tO] f o r some h _> 0 ,

of &

> 0,

a(t) < xo(t) 5 p(t) for a l l x o ( t ) > p ( t ) f o r t E (h,h+E].

s a t i s f y i n g (3.3.1) and such t h a t

t E [O,h], xo(h) = @(h) and S i m i l a r l y t h e r e e x i s t s a s o l u t i o n z o ( t ) s a t i s f y i n g (3.3.1), e x i s t i n g on an i n t e r v a l [O,k+ 61 C [O,tO] f o r some k _> 0, 6

> 0,

such t h a t

zo(h) = a ( h ) Proof:

and

a(t)

xo(0) = @(O),

zo(t)].

t

for

for a l l t

We show t h e existence of

y i e l d s t h e existence of letting

-< Zo(t) -< @ ( t )

zo(t) < a ( t )

E

xo(t)

E [ O J ~ ] ,

[k,k+61.

[ a similar proof

#

F i r s t assume bl

we o b t a i n from (3.3.4)

that

0;

then,

xA(0)

>

Hence t h e r e s u l t holds f o r h = 0.

p'(0).

Let

bl

#

0

and consider a s o l u t i o n s ( t )

s a t i s f y i n g , f o r each i n t e g e r

n,

xn(o) = r a 11'

169

x;(o> = n.

of (3.2.1)

3. TOPOLOGICAL METHODS

> M(B),

For N

there e x i s t s

s

0

E

[O,tO) satisfying %(so) =

and a ( t ) c % ( t ) < B(t), t E (O,so). Assume not; then e i t h e r cy(t) < % ( t ) < @ ( t ) f o r a l l t E (O,tO) o r there B(so)

T < to such t h a t x,(Tn) = a(Tn), a ( t ) < x n ( t ) < n for t E (O,%). I n e i t h e r case, there e x i s t s a

exists a

B(t), tNE [O,tO] such t h a t lXrj(tN)I 5 B. [In the second case, we make use of the f a c t < a r ( t N ) , since cy(t) i s a s t r i c t lower solution. ]

s(5)

Lemma 3.3.1,

since xP;(O)

= N,

ls(t)l-<

f o r t E [O,tNl and we have a contradiction f o r N > M(B). M(B)

Thus f o r some h E [O,t,I,a(t) 5 B(t) for a l l t E [O,hl and %(h) = p(h). Since p ( t ) i s a s t r i c t upper solution, we have

> B'(h). Thus, there e x i s t s an E > 0 such t h a t % ( t ) > p(t), t E (h,h+&]. The proof i s complete. Xp;(h)

U M M A

3.3.3.

by a parameter

simultaneously.

x(t,p)

Let

Let

p.

x(t,p)

open exterior of

U Int

where

t

M

=

C

be continuous i n

t

and

Let C be a simple closed curve i n the

plane and denote by I n t C C

be a family of functions indexed

and x ( t , p )

(t,x)

and E k t C the open i n t e r i o r and

Suppose t h a t f o r each

C.

p

p, (O,x(O,p))

is defined on an i n t e r v a l

E

[O,TM]

( T ,x(Tp,p)) E Ext C. Define y(p) = t where CI CI sup[t > 0 : ( t , x ( t , p ) ) E I n t CI. Then y is bwer semi-

continuous.

If, i n addition, each M c t i o n

x(t,p)

has the

t

t + h ), P' CI l.l then y i s continuous. (Lemma 3.3.3 is another version of the Wazewski method.) additional property t h a t

THEOREM 3.3.1.

B(t)

(t,x(t,p))

E

Ext C

for

E

(t

Assume t h a t there e x i s t s an upper solution

-

and a lower solution a ( % ) of (3.2.1) with cy(t) < B(t)

Let f f o r a l l t E [O,tOl, s a t i s w i n g (3.3.3) and (3.3.4). satisfy Nagumo's condition with respect t o cy and B. Then there is a solution x ( t ) of t h e B" (3.2.l), (3.3.1), (3.3.2) 170

3.3. WAZEWSKI RETRACT METHOD

Proof:

We may assume t h a t (3.3.5)

since we may redefine

g

o u t s i d e of

holds f o r a l l x D

E

R

so t h a t t h e Nagwno’s

condition w i l l hold. that g

We f i r s t prove Theorem 3.3.1 under t h e stronger hypotheses w ( t ) and B(t) a r e s t r i c t lower and upper s o l u t i o n s and

i s l o c a l l y Lipschitzian.

We w i l l then show how these

hypotheses may be weakened. Consider the s e t of s o l u t i o n s of (3.2.1) and (3.3.1) such that a ( 0 ) < x(0) < p(0). These s o l u t i o n s form a one-parameter family of functions a r e connected.

{x(t,p)],

where t h e s e t of values of

Define a map cp:

P

--f

( t p , x ( t p , PI),

t i s defined i n Lemma 3.3.3 i n which C P Jordan curve forming t h e boundary of t h e region x ( t , p) i s continuous i n both t and p since Lipschitz and thus applying Lemma 3.3.3 we have continuous. Thus cp i s continuous and {rp(p)} Combining Lemma 3.3.2 w i t h t h e connectedness of have the existence of v1 and p2, such t h a t

is the

where

Since

CY

and

p

0

are strict solutions

171

D.

g

Now i s locally

y: p -+ t is CL i s connected. {cp(p)}, we

3. TOPOLOGICAL METHODS

The solution x ( t , p o )

i s a solution of BVP (3.2.l),

(3.3.2). We no longer assume From (3.3.6),

g

(3.3.1),

s a t i s f i e s a Lipschitz condition.

there e x i s t s an

>0

E~

such t h a t i f

<

0

E

<

then

There e x i s t s a sequence of l o c a l l y Lipschitzian functions

E gn ( t , x, Y 1Inzl M

2 [O,tO]X R

.

Ign(t,x,y) Pick N

which converge t o

g(t,x,y)

Assume t h a t

- g(t,x,y) I 5

1/n

so large such t h a t

for 1/N

5

n

> N, a

(t,x,y)

[0,t0I X R

2

and

E~

1/< ~ inf[Iatt - g ( t , a , a l ~ ,

Thus, f o r

uniformly on

IB"

-g(t,~,~f)Il,

p w i l l be s t r i c t lower and s t r i c t

and

upper solutions for (3.3.7)

XI'

= gn(t,x,x'),

where Ign(t,x,y)

I 5 h ( IY I )

+

(l/n)*

By o u r previous analysis, t h e r e e x i s t s a function s ( t ) s a t i s f y i n g (3.3.7), (3.3.1), and (3.3.2). From t h e mean value theorem, there e x i s t s a such t h a t

Ix'(T)I

5

K/to

Choose M = M(K)

= K1.

so large t h a t

SMiqq K1

Then, t h e r e e x i s t s

N1,

sds

'K*

such t h a t f o r 172

n

> N1

T

E

[O,tO]

E ~ ,

3.3. WAZEWSKI RETRACT METHOD

s

s ds + (l/nl

h(s) K1

As i n the proof of Theorem 1.4.1,

> K.

we have f o r

t

Ix;(t)l

IM(K)

for

all

Ixn(t)I

IK

for

all t

n

> N1j

E [OJtO].

Since E

[OJtO]J

it follows t h a t Ixi(t)

I I max[lg(tJxJy)I 1x1

Therefore t h e sequence

5 K,

[O,to],

+ 1:

I MI. for n > max(N,N1)

IYI

{xn(t)?,

subsequence which converges i n t h e

C(l)

function i s then a solution of (3.2.1)

norm.

has a

The l i m i t

and s a t i s f i e s (3.3.1)

and (3.3.2). We now remove t h e r e s t r i c t i o n t h a t a ( t ) and B(t) are s t r i c t lower and upper solutions. Let p: [O,tO] -+ (O,tO] satisfy

p"(t)

i = 1,2,

and

-< 0

for a l l t

E

[O,tOl Li(p) 2 0

a,(t) = a ( t )

Define 173

for

i = 1,2,

for

i = 1,2.

for

- EP(t).

3. TOPOLOGICAL METHODS

and l e t

where

6 ( & ) i s t h e modulus o f c o n t i n u i t y of

g(t,x,y)

domain D' = [ ( t , x J y ) : t E [ 0 , t 0 I ,

1x1 5 K, IyI i s defined as before and S > max (M(K), "Pt

5

K

E [0

[O,tOl I p ' ( t ) l ) .

Since

and

Ig(t,c?Jc?')- g(t,ct&Jc?;)l <

y& ( t , a E Ja E' ) E

to1 l a w I >

<

we have

= -2

g ( t , a , a ' ) = B(t,CY&,CYL) E

where

laE(t)-a(t)l + la;(t)-al(t)l

SU%€

Moreover, since

SIJ

on t h e

E

- 6(E).

Hence

a; > a"

aE i s a s t r i c t l o w e r s o l u t i o n o f

Thus

Similarly

BE i s a s t r i c t upper s o l u t i o n of (3.3.8).

Furthermore,

as

-> g ( t , a , a ' ) > gE(t,aE,a;).

E

+O.

g (t,xJy) E

Therefore

converges t o

g (t,x,y) E

g(t,x,y)

uniformu

satisfies Nagumo's condition

and as before t h e r e e x i s t s a s o l u t i o n x E ( t J x , y ) of (3.3.8), (3.3.1),

and (3.3.2).

Moreover, 174

Ex,}

converges i n t h e

C(l)

E

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

The l i m i t i n g f'unction i s then a s o l u t i o n of the BVP (3.2.1), (3.3.1), and (3.3.2). This concludes the proof of Theorem 3.3.1.

norm as

E

+ 0.

REMARK 3.3.1.

The significance of t h i s r e s u l t i s that we can show existence of solutions of t h e Neumann problem, that i s , when a1 = a2 0 i n (3.3.1) and (3.3.2). Observe t h a t except f o r t h i s very important case Theorem 3.3.1 i s a consequence of Lemma 3.3.2 and Theorem 3.2.2. The following example i l l u s t r a t e s Theorem 3.3.1. EXAMPLE 3.3.1.

Assume r1

=

and. satisfies Nagumo's condition.

r2 = 0 Let

and

(3.3 . l o ) then, from (3.3.9)

c > 1/E

t € [O,

11

e x i s t and satisfy

gx

Then the boundary value problem (3.2.1), a solution. To see t h i s l e t Ig(t,O,O)

g: [0,1]xR2 - t R

(3.3.1),

(3.3.2) has

I;

we have

g(t,O,O) -g(t,-C,O) _> CE. Using (3.3.10 ) , we then obtain

g(t,-C,O) < 0. Thus a ( t ) = -C i s a b w e r solution and similarly ~ ( t =) c i s an upper s o l u t i o n f o r (3.2.1), (3.3.1), and (3.3.2). By Theorem 3.3.1, there exists a solution of t h e BVP (3.2.l), (3 03 el), (3 e3.2). 3.4

GENERALIZED DIFFERENTIAL EQUATIONS

I n t h i s section we develop a Wazewski theorem f o r con-

tingent d i f f e r e n t i a l equations and. u t i l i z e it t o prove 175

3. TOPOLOGICAL METHODS

existence theorems f o r BVP's. The following preliminaries w i l l prove usef'ul i n t h e discussion.

Let

(cc(Rn)) be t h e c o l l e c t i o n of a l l

c(Rn)

nonempty compact (compact and convex) subsets of

x

and A,B E c(Rn),

E Rn

and

let

For

Rn.

q(x,B) = i n f [ l l x - b ( ( : b c B]

q(A,B) = sup[q(x,B): x E A].

Then

d(A,B)

I

max[q(B,A),q(A,B)] i s t h e Hausdorff metric on c(Rn) and n c(R ) i s a complete metric space. Let V be a subset of R x R n and denote t h e points of V by

p = (t,x).

continuous (USC) at a

6

>0

such t h a t

If we replace

continuous

> 0,

p E V

i f , f o r each

E

IlQ-pII

<

q(F(Q),F(p))

implies

6

q(F(Q),F(p)) by

d(F(Q),F(p)),

X be an open subset of

Let

be USC.

.

i s upper semi-

F: V + c(Rn)

A mapping

RxRn

there e x i s t s then

and l e t

<

E.

is

F

F: X +cc(Rn)

A s o l u t i o n of t h e generalized d i f f e r e n t i a l equation

(3 4.1)

x ' E F(t,x)

9

i s a function

cp: I + R

n

which is absolutely continuous on

each compact s u b i n t e r v a l of everywhere on For

I

and

cp' ( t ) E F(t,cp(t))

almost

I.

E C X

and

po = (to,xo),

an accumulation point of

at p0 i s D+(E,pO) = [y E Rn: t h e r e e x i s t s {(tn,xn)3 C E, (tn,xn) + (to,xo), tn > to, ( x n - x O ) / ( t n - t O ) + y as n + m I . The negative E,

E

t h e p o s i t i v e contingent of

contingent and contingent of

similar manner. then w r i t e

If

+ D cp(to)

E

E

at

p

0

a r e defined i n a

i s t h e graph of a function

instead of

+ D (E,pO).

cp: I +Rn,

A solution of t h e

contingent equation

(3.4.2)

Dx C F ( t , x )

i s a continuous function

F(t,cp(t))

f o r a l l t E I.

cp: I + R n

such t h a t

A function

176

cp(t)

Dcp(t) C i s a solution

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

of (3.4.1) if and only i f it i s a s o l u t i o n of (3.4.2). The b a s i c theory f o r generalized equations hold; namely, t h e Peano's e x i s t e n c e theorem, e x t e n d a b i l i t y of s o l u t i o n s , and t h e Kamke convergence theorem are true f o r generalized equations, and we shall not expound on t h i s f'urther. Denote t h e maximal i n t e r v a l of existence of a s o l u t i o n D Before s t a t i n g a Wazewski theorem f o r cp' generalized equations, we w i l l need t h e following d e f i n i t i o n .

cp(t)

of (3.4.1) by

DEFDIITION 3.4.1.

A s e t A c X i s ( p o s i t i v e l y ) weakly i n v a r i a n t

with r e s p e c t t o (3.4.1) if f o r each po = ( t O , x o ) E A , t h e r e i s a solution V(t) of ( 3 . 4 . l ) w i t h cp(to) REMARK 3.4.1.

(t,cp(t)) E A

A closed set

v a r i a n t if and only if

LEMMA 3.4.1.

=Xo,

on

D cp

I [tO,m).

i s p o s i t i v e l y weakly in-

E

+ D (E,pO) fl F(pO) f 6

f o r all po

E

E.

are r e l a t i v e l y closed, X with X = E U E2. 1 i s p o s i t i v e l y weakly i n v a r i a n t .

Assume

and E2 1 p o s i t i v e l y weakly i n v a r i a n t subsets of Then H = E fl E2 1 Proof:

Let

two s o l u t i o n s and an Let

a> 0

L(t)

E

po = ( t O , x O ) E H.

qi(t)

By hypotheses, t h e r e e x i s t

of (3.4.1) with (t,cpi(t))

such t h a t

be t h e segment j o i n i n g

E

(pi(to) = xo, Ei

to 5 t < to + a.

for

(t,cpl(t))

i = 1,2,

to

(t,cp,(t)

)

( E n~ E~). Then x ( t ) = a(t)cpl(t) + (1-ct(t))cp2(t), 0 5 a ( t ) ,< 1. Choose a sequence {t,); + t n + t o , and a ( t n ) +ao. Then and l e t

(t,x(t)) E L(t)

n

We now can choose a subsequence

{t,]

t h e l i m i t t h e l e f t s i d e belongs t o s i d e equals

aovl + ( 1 - a o ) v 2

where

177

of

{t,}

Df(H,pO) vi

E

such t h a t i n

and t h e r i g h t

D+cpi(to).

3 . TOPOLOGICAL METHODS

By convexity, D+(H,pO) H

cy

n F(pO) # fl

v + (1-wo)v2 i s i n F(pO) and thus 0 1 and as a r e s u l t , by our previous remark,

i s p o s i t i v e l y weakly i n v a r i a n t .

This concludes t h e proof of

Lemma 3.4.1. The following i d e a s w i l l be needed f o r t h e Wazewski theorem. For

vc

po = ( t O J x O ) E

t h e zone of emission r e l a t i v e t o

X,

is

Ev(po) = [(T,y): y = cp(T), cp(t) with cp(to) = xo and ( t , c p ( t ) ) E V and

21,

v

i s a s o l u t i o n of (3.4.1)

t between to

for a l l

and t h e r i g h t zone of emission r e l a t i v e t o

'

is

V

E;(Po) = [(.JY>: ( T ~ Y )E Ev(Po), >_ t o ] . If A X J Ev(A> = i s s i m i l a r l y defined. U[EV(p): p E A ] and E+(A) V L e t W be a r e l a t i v e l y closed subset of X. For po E W, t h e t r a c e of emission r e l a t i v e t o W i s defined t o be T&~)

(aw

n

n

= $(Po)

(aw

n w)

+

A c w, T ~ ( A )= $(a)

and, f o r

n

w). DEF~ITION

3.4.2.

A point

Q E

aw n w

i s a s t r i c t egress

p o i n t r e l a t i v e t o (3.4.1) i f f o r every s o l u t i o n s u p [ t : (s,cp(s,Q)) E a W ll W, t

<

Q-

6

_< t l < m,

cp(t,Q), c = cp and t h e r e e x i s t s

with ( tn,cp(tn,Q)) E X - W. {t,], tn + c cp We w i l l denote t h e s e t of s t r i c t egress p o i n t s by

a sequence

Finally, a solution

cp(t,p), p E W,

i f t h e r e exists some

( tlY(p(tlJP))

x- w.

tl

E D

cp

n

[t P'

of (3.4.1) m)

S.

"leaves

W,"

such t h a t

Before s t a t i n g t h e Wazewski theorem we w i l l need t h e following lemmas. LEMMA 3.4.2.

p o i n t s of

Let

z C i n t W U S and assume t h a t a l l a r e s t r i c t egress p o i n t s . If a l l

+ + T (2) = T (z) W

s o l u t i o n s through

p C W U S l e a v e W,

then

T+(p)

is

compact. Proof:

We s h a l l show any sequence 178

{$I

C

T+(p)

contains

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

a subsequence which converges t o a p o i n t of frpn(t)] be s o l u t i o n s of (3.4.1) through p

+

Let such t h a t

T (p).

an extension of m e ' s convergence (tn,cpn(tn)) = $. theorem t o contingent equations, t h e r e i s a subsequence {cpk(t)]

of

{cpn(t)} converging t o a s o l u t i o n

of (3.4.1)

cp(t)

with

cp(t ) = x By assumption cp(t) leaves W. Thus P P' ta > t such t h a t ( ta,cp(ta) ) E X - w. there exists P For a l l k s u f f i c i e n t l y l a r g e , ( t a , ( P k ( ta ) ) E X - w , hence

-

t < tk < ta. Choose a subsequence {t,] such t h a t tm+ P tb < ta' Then % = (tm,'pkctm)) + (\,(P(tb))* NOW (tb,cp(tb)) j ~ + ( p >implies f o r m s u f f i c i e n t l y l a r g e , t h a t &m i s not a s t r i c t egress point. This l e a d s t o a c o n t r a d i c t i o n and thus W 3.4.3.

L

Let

a l l points USC map on Z. Proof:

p

as

[pn]

n + m.

that

E

Z

sequence

pn

and

{cpk(t)}

of (3.4.1) with

for each

then t h e r e i s a n

such t h a t

Hence f o r each

q(g,T+(p,))

through

W,

p

Z,

E

~ + ( p >i s compact.

3.4.2,

not USC a t some po E Z, sequence

leave

W U S.

+ T

a r e s t r i c t egress p o i n t s , then

emm ma

BY

i s compact.

be a nonempty subset of

Z

a l l s o l u t i o n s through

+ T (Z)

T+(p)

>

E.

$. of

pn + p o and n, t h e r e i s a

Let

cpn(t)

E

>

If

If

and if i s an

is

T+

and a sub-

0

q(T+(p ),T+(p,)) 2

$

E T

+n

(p,)

such

be a s o l u t i o n of (3.4.1)

By t h e convergencetheoremthere i s asub-

converging t o a s o l u t i o n cp(t)

{cp,(t)]

cp(to) = xo.

By assumption,

A s i n t h e proof of Lemma 3.4.2,

cp(t)

leaves

we can f i n d a subsequence

+

W.

{t,}

of Etk3 such t h a t (tm,~mCt,)) = &m + ($,cp(tb)) E T (Po) which i s a contradiction. This completes t h e proof of Lemma 3.4.2. We now s t a t e and prove t h e Wazewski theorem which w i l l

then be used t o o b t a i n t h e existence of s o l u t i o n s of BW's 179

E

3. TOPOLOGICAL METHODS

associated with THEOR334

points of

(3.4.1). Z

Let

3.4.1.

i n t W U S be connected.

C

T+(Z) a r e s t r i c t egress points and i f po

not connected, then t h e r e e x i s t s

Proof:

Assume not; then f o r any

depends on

and t h e r e exists

W

leaves

such t h a t

cp)

( t ,cp(t

[ t , t 1. P Q compact f o r each p E Z. (t,cp(t,p))

E

on

W

Q

Q”

we w i l l show

w

E

on

D cp

p

E

E

T+(p) c aW

Q

))

By Lemma

T+(Z) i s

and a solution

Z

(3.4.1) such t h a t (t,cp(t,p,))

cp(t,pO) of

cp(t,p)

E

If all

E

n

[tO,m).

every solution

Z,

aw n w

nW

(Q

and

3.4.2, T+(p) i s

is connected. If i s a s t r i c t egress point. Clearly, T+(p) i s then connected. Let V = i n t W, p1 E V fl Z, -Iand assume t h a t T (p,) = C , U C2 is a separation where CY For each

n

p E Z

C2

S,

p E Z,

then

p

+

T (p)

E

T+(p)

p

and

a r e nonempty d i s j o i n t compact s e t s . For a solution

cp(t,p)

of

(3.4.1) with p

E

let

V,

I = D fl [tp,m) = [ t w) be t h e r i g h t maximal i n t e r v a l of c p c p P’ existence of cp(t,p) r e l a t i v e t o X and l e t J = [ t p , y ) be cp t h e r i g h t maximal i n t e r v a l of existence of cp(t,p) r e l a t i v e t o

V.

Let

to

V,

o(p) and l e t

denote t h e r i g h t t r a j e c t o r y o f u(A,B)

-

= inf[((a bll: a

E

cp(t,p)

A, b E B]

,

relative where

a r e a r b i t r a r y sets.

A,B

El = [p E V: t h e r e e x i s t s a solution cp(t,p) I =J or t h e r e exists a solution cp(t,p) such t h a t

Define with

c p c p

with a(O(p),C1) 5 o(o(p),C2)]. Define E2 s i m i l a r l y Icp # Jcp with t h e inequality reversed. It follows t h a t El, Eg a r e closed r e l a t i v e t o V, a r e p o s i t i v e l y w e a k l y invariant, such t h a t p 1

E

E1

n E ~ . ~y emm ma 3.4.1, q(t,pl)

u E2

t h e r e i s a solution

of (3.4.1) such -“hat (t,cp(t,p,)) However.,

V = El

E, must leave W. 180

E

n E*

on J

and rp(t,p,)

cp’

Hence (y,cp(y,pl))

E

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

+

Let T (p,) = C1 U C2. Assume t h a t ( y,cp(y,pl)) E C1. be a sequence of points on O(pl) such t h a t % + ( y , p ( y , p l ) ) ~

g E E2,

Since

C1.

(3.4.1) Since

such t h a t

t h e r e e x i s t s a solution

{

cp(t,P,>,

tPl 5 t

cp(t,g),

tB, c t <

i s a solution of (3.4.1) through through

5

wn,

of

$(y,pl) = (p(y,pl)

s u f f i c i e n t l y large,

+ $(t,pl)

$k(t,pl)

+

(y,cp(y,pl)) E C1 C T (p1) i s a contradiction. Thus Lemma 3.4.3,

$,(t,p,)

uniformly on

Jr(t,pl)

and a subsequence

{ q n ( t ) } such t h a t f o r any compact i n t e r v a l

all k

must leave

eh,

This implies t h a t there e x i s t s a solution (3.4.1) with

I

I

k

+ m.

of

{qk(t)l C

D

i s defined on as

n.

tan>

p1 E Z, $n (t,pl) ( yn,cp(yn, ) E C2.

and hence

C2

of

> 0(@(%),C2) f o r each CJ(O(%),C,) -

$*(t,P1) =

W

cp(t,$)

JI

I

and and

However, then

i s not a s t r i c t egress point, which T+(p,) i s connected. Finally, from

i s an USC map on 2 from which it follows

T'

This i s a contradiction t o t h e assumption

T+(Z)

i s connected.

T+(Z)

i s not connected.

po E Z

and a solution

(t,cp(t,po)) E W on of Theorem 3.4.1.

I

Cp

Hence we conclude t h a t there e x i s t s Cp(t,po)

of (3.4.1)

= DCp fl [tO,w).

such t h a t

This concludes t h e proof

An a l t e r n a t e way of s t a t i n g Theorem 3.4.1 i s i n t e r n s of retracts.

Let

A, B be subsets of

Rn+'

with

t h e r e e x i s t s an USC mapping

G: A -+ c(Rn+l)

i s connected f o r a l l x E A

and x E G(x)

then

B

i s a r e t r a c t of

THE!ORE!M 3.4.2.

t h a t a l l points of

Let

B C A.

such t h a t

If

G(x) C B

f o r a l l x E B,

A.

Z

be a subset of

int W U S

T+(Z) a r e s t r i c t egress points.

181

If

such

3 . TOPOLOGICAL METHODS

Z

n

exists

+

i s a r e t r a c t of

T+(Z)

po E Z,

T (Z), but not of

such t h a t

(t,Cp(t,p,))

E

Z, t h e n t h e r e W for a l l t E D n Cp

[to,..). If t h e conclusion does not hold, then f o r a l l

Proof:

Z, every s o l u t i o n cp(t,p) of (3.4.1) leaves W, and + i s USC on Z. Let H: T+(Z) + T+(Z) fl Z be a rehence T + + t r a c t i o n o f T (Z) onto T (Z) n Z which i s assumed t o e x i s t . "hen H: T+ i s a r e t r a c t i o n of z onto T+(z> n Z. "his i s a p

E

c o n t r a d i c t i o n and concludes t h e proof o f Theorem 3.4.2. "he preceding r e s u l t s remain v a l i d i f

open subset of

X,

[a,m) xRn, W

X

i s a relatively

i s a r e l a t i v e l y closed subset of

and Z i s a connected subset of

i n t W U S.

We now apply

Theorems 3.4.1 and 3.4.2 t o BVP's a s s o c i a t e d with second-order contingent equations. Consider t h e generalized second-order d i f f e r e n t i a l equation x" E G ( t , x , x t ) ,

(3.4.3) where

G:

2 [a,b] X R

--3

cc(R)

extension, we may assume

i s upper semicontinuous (USC). By on X = [a,m)xR 2 Letting

.

i s USC

G

= Y2' H(tJy) = (Y2JG(tJY1JY2))J = Y1J 2 we see H: X --3cc(R ) i s USC and

(3.4.4)

A mction

s o l u t i o n of (3.4.3) on (3.4.5) for a l l t

I

Dxt(t) E

DEFINITION

I

and

C

C

[a,..)

x ( t > E c ( ~ ) ( I ) is a

i f and only i f

G(t,x(t),x'(t))

Dxt(t)

3.4.3.

y = (Y1'Yp)J

H(tJy)

Y'

i s equivalent t o (3.4.3).

where

#$

f o r a11 t E I.

The functions

+ ( t ) J cp(t)

are called

s t r i c t upper, lower s o l u t i o n s f o r (3.4.3)J r e s p e c t i v e l y , if, +(t), q ( t ) E C(l)(I)J I

C

[a,..) 182

and if

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

We w i l l assume t h a t

Note t h a t surface of

W

O(t) < $ ( t )

t

for a l l

i s a r e l a t i v e l y closed subset of i s the s e t

W

= [(t,x,x'):

S

1 and the lower surface of

$ ( t ) , x ' E R]

W

[ ( t J x J x ' ) : t E [aJb], x = cp(t), x ' E R]. connected subset of

W fl [ t = a ] such t h a t

S1 a t a single p i n t of single point of

A2.

and

A1

t

E

!he upper

X.

[a,b], x =

E

i s the s e t Let Z1

S2 =

be a

Z1

intersects

intersects

Z1

Define

I.

S2

at a

We now present a theorem which i s i n the

same s p i r i t as Theorem 3.2.1. THEOREM 3.4.3.

If

$(t)

lower solutions of (3.4.3), (i) (ii)

and

cp(t)

are s t r i c t upper and

then e i t h e r

T+(Z,) fl [ t = b ] # there i s a solution

or

$J

x(t)

of (3.4.3)

(&,X(a)Jx'(&)) z l J cp(t) 5 x(t) 1. $(t) On + where [aJw+) (w+ < b) I x ' ( t ) l - + m as t -+w maximal i n t e r v a l o f existence of Proof:

Assume

+

such t h a t

(t,x(t),x'(t>) E

for

w

t

E

) J and i s the r i g h t

x(t).

T (Z,) il [ t = b ] =

solution of (3.4.3)

such t h a t

+

8.

Let

x(t)

be a

( a , x ( a ) , x ' ( a ) ) E ZIJ [a,t,]

183

and

( t O J x ( t O ) ' x ' ( t O )E) S 1.

3 . TOPOLOGICAL METHODS

Then x ( t o ) = +(to)

and

5

x(t)

$(t)

t

for

E

[a,tol,

Similarly s o x ' ( t o ) 2 + ' ( t o ) . Hence ( t O , x ( t O ) , x l( t o ) ) E A1. a solution from Z can intersect the lower surface S2 only 1 + a t points of A2. Thus T (Z,) C A 1 U A2. Observe also T+(Z1) since A1 fl Z1 Let

n A1 # 9

##

and

and A2 flZ

T+(Z,)

n

A2

##

# #.

1+ i s a s t r i c t egress point. We now show any point of T (Z,) + Q = (tO,xO,x;)) E A1 n T (Z,). Assume Q i s not a s t r i c t

egress point; then there e x i s t s an i n t e r v a l and, a solution x ( t ) + ( t ) on

[ t , , ~ ] , to < T, Q such t h a t x ( t ) <

emanating from

[ t , , ~ ] . Clearly,

x'(to) = +'(to).

Let

-

and consider the sequence

-

-

tn + t o

{ [XI(t,) x ' (t,)/(t, t o ) ] . Since Dx' ( t o ) C G ( t O y x ( t O ) , x T ( t O )and ) since G(tO,x(tO),xt( t o ) ) i s compact, the sequence i s bounded. Thus there e x i s t s a subsequence {t,] such t h a t

=

G(t0>+(t0)'+l(t0)).

-

cause

+(t)

-

{ [ + I (t,) +' ( t o ) ]/(t, to)}. Bei s a s t r i c t upper solution, t h i s sequence contains

Consider now the sequence

a subsequence converging t o a point i n there e x i s t s a subsequence +'(tj)

for

8 E (t,,~]

implies

j

of

s u f f i c i e n t l y large.

such t h a t

x(e) >

{t,]

$(e),

shown t h a t any point of

x'(t) > $!(t)

{t,] such t h a t x ' ( t . ) > J Hence there e x i s t s

n

T+(Z,)

t

for

a contradiction.

A2

E

( t o , @ ) . This

Similarly, it can be

i s a s t r i c t egress point.

We can now apply Theorem 3.4.1because

184

Therefore,

(-m,y).

a l l points of

3.4. GENERALIZED DIFFERENTIAL EQUATIONS

T + ( z ~ ) a r e s t r i c t egress p o i n t s and Hence t h e r e i s a s o l u t i o n x ( t )

T+(z,)

of (3.4.3)

i s not connected. such t h a t

( a , x ( a ) , x l ( a ) ) E Z1 and cp(t) < x(t) < $ ( t ) on [a,w+) with + + w < - b. Furthermore I x l ( t ) l - + m as t + w , f o r i f not,

+

fl [ t = b ] # $.

T (Z,)

holds.

Hence i f ( i ) does not hold, then ( i i )

This concludes t h e proof o f Theorem 3.4.3. The proof o f Theorem 3.4.3

REMARK 3.4.2.

than t h a t of Theorem 3.2.1. Theorem 3.4.3

i s different

We could have, however, proved

i n a manner similar t o t h a t of Theorem 3.2.1.

This follows from t h e fact t h a t it can be shown t h a t t h e r e + exists a connected subset of T (Z ) i n W fl [ t = b ] which 1 A

and A2. 1 We now extend t h e concept of Nagumo's condition t o second-

intersects

order contingent equations. DEFINITION 3.4.4.

The function

G:

[a,b] X R

satisfies Nagumo's condition r e l a t i v e t o W p o s i t i v e function

G(t,x,x')] < h ( Ix'

h ( s ) E C([O,m))

such t h a t

I)

E

for

(t,x,x')

W

and

2

+cc(R)

if there exists a

max[lzl : z E /" s d s / h ( s ) = +m.

The following lemma can be proved s i m i l a r l y t o t h a t of Lemma 1.4.1.

s a t i s f i e s Nagumo's condition on [a,c], then, given an i n t e r v a l [ a , c ] , a < c < b, t h e r e exists an and i f

N > 0

cp(t)

G(t,x,x')

such t h a t f o r any s o l u t i o n x ( t )

5

x(t)

5

q(t)

EXERCISE 3.4.1. Let

Z2

on

[a,c],

then

of (3.4.3)

Ix'(t)l
with on

[a,c].

Prove Lemma 3.4.4.

be a subset of

W fl [ t = b ]

185

so that t h e r e exists

3. TOPOLOGICAL METHODS

a separation W fl [ t = b ]

n

and A2 El f o r a BVP.

[t =b 1

THEOREM 3.4.4.

C

- Z2

= El U E2

with

A1 fl [ t = b ] C

We now state an existence theorem

E2.

If t h e r e exist s t r i c t upper and lower

solutions f o r (3.4.3)

on

and if G(t,x,x')

[a,b],

Nagumo's condition with respect t o

satisfies

then there i s a solution

W,

o f (3.4.3) such t h a t q ( t ) < x(t) < Jr(t) on satisfies t h e boundary conditions (a,x(a),x' ( a ) )

x(t)

(b,x(b),x'(b) Proof: immediate.

)

If T+(Zl) fl Z2 Assume then t h a t

+

C

S1 U S2

A l l points of

s t r i c t egress points. Q E

A1,

and

Z,

# #, then t h e conclusion T+ (Z,) n Z2 = 8. Then

is

-

u

(W fl [ t = b ] Z2),

and as i n t h e proof of Theorem 3.4.3,

+ T (Z,) n

E

and

Z2'

T (2,)

i = 1,2.

[a,b]

El U E2

C

W

T+(Z,)

n

[t =b]

fl Si C A i

for

are obviously

This together with t h e f a c t , t h a t

i = 1,2,

with

point implies t h a t a l l points of

t < b,

Q

"+(Zl)

is a s t r i c t egress are s t r i c t egress

points (see t h e proof of Theorem 3.4.3). Since

+

(A, U El) U (A2 U E2) and since El U E2 it follows t h a t @(Z,) fl Z1 W fl [ t = b ] Z2, T+(Zl). However, T+(Zl) n Z1 i s not a r e t r a c t

T (Z,) i s a separation o f

i s a r e t r a c t of

C

-

x ( t ) of (3.4.3) with (a,x(a),xl(a)) E Z1 such t h a t ( t , x ( t ) , x l ( t ) ) E W on i t s maximal i n t e r v a l o f existence [t,,w) r e l a t i v e t o X. This means w < - b and I x l ( t ) I 4 m as t +w-. By Lemma + 3.4.4 t h i s i s impossible. Therefore T (Z,) fl Z2 # @ and t h i s concludes t h e proof o f Theorem 3.4.4.

of

3.5

Z1.

By Theorem 3.4.2,

t h e r e e x i s t s a solution

DEPENDENCE OF SOLUTIONS ON BOUNDARY DATA This section u t i l i z e s t h e topological properties of

186

35. DEPENDENCE OF SOLUTIONS ON BOUNDARY DATA

s o l u t i o n funnels t o deduce r e s u l t s on t h e continuous dependence on boundary values of s o l u t i o n s of BVP's f o r systems o f d i f f e r e n t i a l equations with generalized boundary conditions. Let

I be an open i n t e r v a l i n

and

We w i l l consider t h e two point BVP

f = (fl,f2).

(3 5.1)

x' = f ( t , x ) , g ( x ( a ) ) = r,

(3.5.2) where

R, x = (x1,x2),

2

f E C [ I x R ,R

2

1,

h(x(b)) = q

(a,b

I),

E

2 2 g E C [ R , R ] , h E C[R , R ] .

We s h a l l use some of t h e notions dealing with compact sets t h a t were developed i n t h e preceding s e c t i o n . we introduce s i m i l a r n o t a t i o n h e r e . int A

(a)denote t h e

If

A

For completeness

is a s e t , l e t the

i n t e r i o r (boundary) o f

A

and l e t

[a]

be the one-point s e t containing a . 2 Let c ( R ) denote t h e f a m i l y of a l l compact, nonempty 2 subsets of R Let q(x,B) = inf[llx-y(l: y E B], q(A,B) =

.

A], d(A,B) = max ( q(A,B),q(B,A)). When A C R2 and E > 0 l e t N(A,&) = [x: q(x,A)

sup[q(x,B): x A,B E R2 f o r

E

w i l l be denoted by N(x, E). For

X

C

R

2

,

t h e mapping

2 F: X + c ( R )

upper semicontinuous (continuous) a t

x

E

x

5

i s s a i d t o be i f y + x implies

q(F(y), F ( 4 ) -0. As we have observed, (3.5.1) generates a mapping 2

R XIXI i n t o t h e f a m i l y of a l l closed subsets of

R2

F

of

defined

bY F(x,tO,tl) = [y: y = x(t,), x = x ( t ) i s a s o l u t i o n of (3.5.1), x(to) = This i s p r e c i s e l y t h e cross s e c t i o n a t funnel through The mapping

tl

XI.

of t h e i n t e g r a l

(to,x). F has t h e following well-known p r o p e r t i e s :

187

E)

3. TOPOLOGICAL METHODS

(I) F(A,t 0 J t1)

= UIF(xJtO,tl): x E A ]

and connected) provided (3.5.1), (11)

i s a continuum (compact

i s a continuum and a l l solutions of

A

x(tO) = yJ y E A

e x i s t on

[tOJtl].

I f every solution of (3.5.1) through

( t O J x O )E

Q C

IxR2

e x i s t on

A x{to} C Q

and

q(B,A)

[ t o - h , to+ h ] , then given a compact s e t E > 0 t h e r e a r e 6 > 0 and q > 0 such t h a t

<

6,

Is - t o [< 6 implies q(F(BJsJtl)J F(AJtOJtl)) < EJ and

Itl- t21 < q

implies

~ ( F ( A J ~ ~~ J( ~ J~t )o JJ t z<) )E J provided

It2- t oI

BX{S}

C

Q,

t

SJ

5 h.

Let

and l e t

t

1’ 2

E

IJ

IS-tll

5 h,

Ito-t,I

x o ( t ) be a solution of (3.5.1) defined on [ a J b ] C I 2 0 be a subset of [ a J b ] x R The BVP (3.5.1)J (3.5.2)

.

i s s a i d t o be unique r e l a t i v e t o x o ( t ) , [ a ~ b ] and ~ any solution for

t

5 hJ

y(t)

of (3.5.1),

t h e conditions

0,

if for

(t,y(t)) E Q

[CJ~]

g(Y(c)) = g(xo(c))J h(Y(d)) = h(xo(d)) imply y ( t ) = x o ( t > on [CJd]. We a r e now able t o s t a t e a continuous dependence r e s u l t . 6

The proof of t h i s w i l l follow a f t e r a s e r i e s of lemmas.

THEOREM

(i)

3.5.1.

For a l l

f J g, h

Let

r

E

s a t i s f y the conditions:

t h e curves

Rj

K(r) = [x: g(x) = r ] ,

H(r) = [x: h(x) = r ] a r e simple arcs separating R2; ( i i ) For a l l r, q E R, K(r) n H(q) i s a one-point s e t . Let

x = x o ( t ) be a solution of t h e BVP (3.5.1)J and

(3.5.3)

g(x(o)) =

r

~

J

188

h ( x ( l ) ) = 91’

35. DEPENDENCE OF SOLUTIONS ON BOUNDARY DATA

Let t h e BVP (3.5.1),

(3.5.2) be unique r e l a t i v e t o

[0,1], and some open s e t

[(t,x): x = xo( t) , t

Q E [0,1] xR2

xo(t), containing t h e a r c

[0,111.

E

such Then there i s a neighborhood U of (O,l,r o’91) t h a t f o r a l l (a,b,r,q) E U, t h e BVP (3.5.1), (3.5.2) has a solution

= x(t,a,b,r,q)

x

t

uniformly i n

satisfying

If uniqueness of t h e

--f

xo(t)

( a , b , r , q ) + (O,l,rO,ql).

[0,1] as

E

x(t,a,b,r,q)

BVP i n t h e “usual sense” i s assumed,

we obtain immediately t h e following corollary. COROLLARY 3.5.1.

Assume ( i ) and ( i i ) hold and t h a t t h e

solutions of the BVP (3.5.1), a r e unique.

If the solution of t h e BVP (3.5.1),

then t h e BVP (3.5.1), for a l l and

(3.5.2), whenever they e x i s t ,

(a,b,r,q)

(3.5.2) has a s o l u t i o n x = x ( t , a , b , r , q ) i n a c e r t a i n neighborhood U of (O,l,ro,ql)

i s continuous on

x(t,a,b,r,q)

I n order t o prove Theorem 3.5.1, liminary r e s u l t s .

(3.5.3) e x i s t s ,

Choose

>0

&

[0,1] X U . we w i l l need some pre-

such t h a t t h e s e t

[(t,x):

Ilx-xo(t)II 5 2&, t E [0,1]] i s i n Q. L e t h > 0 be t h e number with t h e property t h a t if s E [0,11, l l y - x o ( s ) / / 5 E, then every solution

x = x(t)

[s - h , s + h l

e x i s t s on

of (3.5.1) s a t i s f y i n g

and s a t i s f i e s

t h i s interval. Choose a sequence of numbers

ti+l - t . < h

such t h a t

1

5

2& on

< .-.< tm ..,m - 1, 1and l e t

0 = to < t

i = 0,.

if

Ilx(t) - x o ( t ) l l

x(s) = y

= 1

xi - x0 ( t 1 . ) , Ni = N(xi,&), ri = g(xi), q r = h(xi), Ui = [x: h ( x ) < q r ] , Vi = [x: h(x) > qy]. By the choice of

then

F

h,

if

A C N ( x ~ ( s ) , E ) ,0

s a t i s f i e s (I) and (11) and

For

A

C

Ni - 1, write

sets

Si, i = 0 , .

K(ro)

n No

..,m - 1,

containing x

0’

Fi(A) by

So

‘i+l

189

F(A,s,t) c N ( x o ( t ) , 2 & ) . F(A,t i-1’t i ).

Define the

as the component of as t h e component of

3. TOPOLOGICAL METHODS

n

(S ) Ni+l containing xi+l. Fi+li compact, it follows from (I) t h a t

Since

Si

a r e compact.

uniqueness condition on the BVP (3.5.1), (305.4)

Fi+l (Si)

LEMMA 3.5.1.

n H(qT+l) Let

Ao, Bo

=

=

(3.5.2)

Exi+,l,

=

and Ni

So

i =

satisfy

A.

are

The

implies t h a t

o , i , . ..,m

U Bo = K ( r o )

- 1. n

n ai,

&To,

F ~ ( A ~ ul, ) F ~ ( B ~ Ivl. Let Ai = ~ ~ ( s ~ ui , ~ > Bi = Fi(Siml) fl Vi n bi. Then given T E ( t l , l ] there e x i s t s a S

of

>0

>0

q

and

n

such t h a t f o r every solution

(3.5.1) defined on

Ix(t) - x o ( t ) l

5

x(p) E N(Ai,q)

[O,T] U [p,Tl,

t

x = x(t)

t h e conditions

[o,Tl U [p,Tl; l p - t i l < q, [or x(p) E N(Bi,q)l f o r some ti E LO,")

2~

for

E

bPl-Y

(3.5.5)

h(x(t)) c h(xo(t))

on

n

[t,Tl

or

h(x(t)

> h(xo(t)) + 8

[p,Tl.

Proof: n = 1,2

-8

,...,

Assume t h e r e s u l t i s not true.

Let

yJt),

be a sequence of solutions of (3.5.1)

-

such t h a t

Yn(Pn) E N(Ai,l/n), h(Yn(Sn>)2 h(xo(s,)) ( l / n l , where Ipn-til c l/n, sn E [pn,T] [t,TI. We may assume sn + s 0 and y n ( t ) + y o ( t ) uniformly on [ O , T I .

n

Uij h(Yo(So)) 2 h(XO(SO)), so t h i s implies t h a t t h e r e i s a solution z ( t ) s a t i s f y i n g Hence

yo(ti)

z(0) E K ( r o ) ,

ro,tii,

w(t) =

E

Ai

z ( t i ) = yo(ti).

Y,W

on

Let

w(t) = z ( t )

> ti;

on

[ti,sOi'

we have g ( w ( 0 ) ) = g(xo(o)), h(w(ti)) h ( xo(ti)), h(w(so)) 2 h(xo(so)), contradicting t h e uniqueness. This completes the proof of Lemma IZMMk 3.5.2. 8i(Ei+l)

3.5.1.

For a given

such t h a t if

Is

E

~

- t i / < Ei, 190

>+ 0 ~ t h e r e

is a

1s -ti+ll 5 h,

tji =

3.5. DEPENDENCE OF SOLUTIONS ON BOUNDARY DATA

D c N(siJsi) n N ( X ~ ( S ) , E ) , then a l l components of F(DysJti+l) which i n t e r s e c t H(qi+l) are i n N(Si+l, Ei+l). Ni+l

*

Proof: E ~ + ~ 0, >

Since S = F (S ) n Ni+l i+l i t h e r e is a decomposition of

2

Let

D1(6), N(Fi+l(Si)

-

be t h e union of a l l components o f

>0

6

i n t o compact s e t s

such t h a t

X , X ,S = X1 U X2 1

is compact f o r any S

S , 6 ) fl Ni+l

which i n t e r s e c t

and l e t

N(XlJ6),

Then by (3.5.6) D 2 ( 6 ) = ( N(Fi+l(Si) -S,6) n Ni+l) -D,(S). and (3.5.7), t h e sets Ck(6) = N($,6) u D k ( 6 ) (k = 1 J 2 ) satisfy

provided

6

is s u f f i c i e n t l y small.

Applying t h e semicontinuity of such t h a t if D CN(Si,6) IS-ti+l

I -< h,

F,

n N(xO(S),E),

choose

6. = b . ( E 1

I t i - s l < 6i,

1

)

i+l

then F(ti+lJSJD)

N ( F i + l ( s i ) ~6 ) -

The a s s e r t i o n follows immediately with t h e use of (3.5.8) and t h e formula N ( F ~ + ~ ( S ~ )n, & N ) ~ =+ ~ ~ ~ ( u 6~ )~ ( 6 ) . We a r e now i n a p o s i t i o n t o prove Theorem 3.5.1. Pmof o f Theorem 3.5.1: of t h e s e t

K(r)

n

N(xo(a),

Let E)

191

S O ( r , a ) be t h e component

which i n t e r s e c t s t h e sets

n

3 . TOPOLOGICAL METHODS

> h(xo(a))3.

cx: h ( x ) < h(Xo(a))3, cx: h ( x ) Let

Si(r,a),

i = l,...,m-l,

denote t h e component of

sim1,ti) fl Nij where

F(Si(r,a), satisfying Set

so = a, s j =

# 8,

s i ( r , a ) n ui C(r,a,b) = F(S,

S i ( r J a ) n vi # l ( r , a ) , tm,lJb ) .

t j for

j

fl.

2

1,

prove t h e

TO

theorem, it i s enough t o show t h a t

# fl

(3-5-9)

C ( r ~ a ~ b )H(q)

where

i s a c e r t a i n neighborhood of

U

I n f a c t , (3.5.9) x(tjajb,rjq) Since of

Of

for

E

UJ

(O,l,rO,ql).

implies t h e existence of a s o l u t i o n

t h e BvP (3.5.1)j

x(tiJaJbJr,q)

(aJbJr,q)

E

Ni

for

(3.5.2)

for

(ajbjrjq) E

...,m - 1,

i = 1,

by t h e choice

ti’ I x ( t , a , b , r , q ) - x ( t ) 1 < 2~ on some i n t e r v a l 0

[0,1]. Thus, t h e family

containing

i s equicontinuous on ness of

xo(t),

[c,d].

u.

[c,d]

{x(t,a,b,r,q)},

(a,b,r,q) E U

By Ascoli‘s theorem and t h e unique-

we deduce t h e uniform convergence on

[0,1] of

x(t,aJbJrJq) as ( a J b > r l q ) -3 (OYlYr0Jq1)* I n o r d e r t o show t h a t (3.5.9) i s t r u e , we f i r s t prove t h a t a l l sets enough.

Si(r,a)

exist if

Ir

For t h i s , define constants

- ro I

ai > 0,

N ( K ( ~ ~ ) a, o ) n 3 ~ c 0 N(AO

(3.5.10 )

N ( F ~ ( sl )~J ~ i ) n

a~~ C N ( A u~ B ~ , Q )

(3*5=11)

am-l>

Em-l =

E~ =

and

u

for

Ai, F(.,s

duct ion. Let (3.5.12)

>0

a r e small by

BOJ~),

for

i = lJ...,m-lJ

min(aiJ6i(~i+l)) if

where

E~

la I

i = m-2,...,0,

are as i n Lemma 3.5.1 and 6 i ( * ) i s defined ti+l) a s i n Lemma 3.5.2. We now proceed by ini’

BiJ q

.4

>0

be so chosen t h a t SO(r,a)

So(r,a)

N(SO,EO),

192

e x i s t s and where

3.5. DEPENDENCE OF SOLUTIONS ON BOUNDARY DATA

(3*5*l3)

so(rJa>

N(A~Jfl)

#

So(r,a)

#J

N(B~J?)

##

-

la I 5 a, Ir ro I 5 a. The p o s s i b i l i t y of such an R follows from (i), (ii),and the c o n t i n u i t y of g, h, and xo.

for a l l

kt

k =

min(a,eo,q). If la1 5 k, Ir-rol 5 k, then e x i s t . To see t h i s , observe that i f Si(r,a)

a l l Si(r,a) e x i s t s and i f it s a t i s f i e s the condition

(3.5.14),

Si(r,a)

N(SiJ~i),

and the condition t h a t

there are solutions ui(t), vi(t) (3.5.1) such t h a t u i ( s . ) , vi(s.) J J f o r j = O,...,i; ui(sh) E N ( % , ? ) ,

(3.5.151,

vi(sk) s

then,

Si+l(r,a)

j

=

t

E

j

1

J

f o r some h, k (so = a,

N(BkJq)

for

of f S.(r,a)

-< j -<

i),

e x i s t s and s a t i s f i e s (3.5.14)i+1

and

(3 5*15)i+1I n f a c t (3.5.15)i and Lemma 3.5.1 imply t h a t F ~ + ~ ( u ~ ( ~ ~ ) ) J ); hence a t least one ( v . ( t . ) ) a r e separated by H(q*i+l Fi+l1 1 Fi+l(Si(r,a)) n N ~ +i n t~e r s e c t s u i+l and Vi+l. By Lemma 3.5.2, Si+l(rJa) C N(Si+lJ~i+l), t h a t is, ( 3 . 5 ~ 4 ) , + holds. ~ To show (3.5.15)i+1 holds, choose any point ui+l i n S i + l ( r J a ) ui+l &i+lJi f t h i s s e t is nonempty; otherwise (u ( t ) ) ( v ~ +is~ chosen s i m i l a r l y ) . If l e t ui+l Fi+li i Fi+l(Si(r,a)) n Ui+l n aNi+l # # J then by (3.5.10), Ai+l # compnent of

"

$ 0

Hence by (3.5.14)i+l, (3.5.11), and (3.5.10), ui+l E N(Ai+l,q). Thus i n every case, t h e r e is a s o l u t i o n ui+l(t) through ( t i + l ' u i + l ) satisfying (3.5.15)i+1. By (3.5012) and (3.5.13)J So(',&) e x i s t s and s a t i s f i e s (3.5.14),, (3.5.15), for a l l \a1 5 k, and Ir-rol 5 k. Hence, by induction, Si(r,a) e x i s t s f o r a l l i > 0. 193

3. TOPOLOGICAL METHODS

To prove (3.5.9), observe t h a t the existence of (r,a) implies the existence of C(r,a,t) f o r a l l

'm- I

14 5 k,

+hl. By property Ir-rol 5 k, and t E [trn,& trn-l c ( r , a , t ) is continuous i n t on 1' tm- 1 + hl. Thus (3.5.15)m-1 and Lemma 3.5.1 imply t h a t (II),

(3 5 16) I

E

C(r,a,b)] < h(xo(b))

max{h(x): x

E

C(r,a,b)] > h(xo(b)) + 8

-

1. 5 k,

hold f o r

-

minCh(x): x

8,

-

Ir r o \ ,< k, Ib 11 5 kl, where kl > 0 is sufficiently small. Since C(r,a,b) is connected, then ( i ) and (3.5.16) i m p l y t h a t (3.5.9) holds, thus completing the proof o f Theorem 3.5.1.

Equation (3.5.1) with the boundary condition (3 -5-17) m a ~ rbe

g(x(b))

&(a>) = r,

=

9

considered as a special case of the BVP (3,5,1), (3.5.2).

EXERCISE 3.5.1.

State and prove a theorem similar t o Beorem 3.5.1 f o r the BVP (3.5.1), (3.5.17). We now consider a special case of Theorem 3.5.1,

fl(t,xl,x2)

= x2

and g and h are linear.

THEOREM 3.5.2. (i)

(t,y)

E

Assume t h a t

x1 < x2 implies f(t,xl,y)

[a,blxR;

(ii)

If(t,x,y)

- f(t,x,z) I

5 f(t,x2,y)

5 Mly - zI on

( i i i ) aO,al,bo,bl 2 0, a. + bo >

bo

when

+ bl > 0.

0,

f o r a=

[a,bl xR2;

a. + a1 > 0,

Then the BVP x" = f ( t , x , x ' ) ,

aox(a) - a 1x l ( a ) = r,

194

box@) + blxl(b) = q,

3.6. NOTES AND COMMENTS

has a unique solution x'(t,r,q)

x(t,r,q)

a r e continuous on

EXERCISE 3.5.2.

Hint:

f o r any 2 [a,b]xR

.

r, q

and

x(t,r,q),

Prove Theorem 3.5.2.

Show t h a t

e x i s t s and i s unique by

x(t,r,q)

a b + albo 0 1 implies condition ( i i ) of Theorem 3.5.1 holds.

u t i l i z i n g ( i ) and ( i i ) .

By ( i i i )

> 0.

This

Apply f i n a l l y

Corollary 3.5.1. In t h e s p i r i t of Exercise 3.5.1, we s t a t e the following result.

THEOREM 3.5.3.

Let

f: I X R

2

+R

be continuous,

Let

x o ( t ) be a solution of t h e BVP (3.5.18)

X"

= f(t,XJX'),

corresponding t o

is a

6

>0

f(t,XJx')

x(b) = 9,

(a,b,r,q) = (O,l,roJql).

such t h a t i f

with

X(a) = r, x(t)

a,b E 1

Assume t h a t there

i s a solution of

x" =

x ( a ) = xo(a), x(b) = xo(b), a,b E [0,1

I x ( t ) - x o ( t ) l < 6, I x l ( t ) - x l ) ( t ) l x ( t ) E x o ( t ) on [a,b].

< 6 on

Then the BVP (3.5.18) has a solution

[a,b],

t 9

then

x(t,a,b,r,q

EXERCISE 3.5.3. Prove Theorem 3.5.3 by applying Theorem 3.5.1 t o the s p e c i a l case i n which g(x,y) E h(x,y) = x and

fl(t,X,Y)

3.6

= Y.

NOTES AND C O ~ ~ T S Theorems 3.1.1 and 3.1.2 of Section 3.1 are taken from

Bebernes and Wilhelmsen [6] , while the subsequent discussion 195

3. TOPOLOGICAL METHODS

dealing w i t h Wazewski's method is based on Jackson and Klaasen

[3] who t r e a t a more general problem not demanding uniqueness. Theorems 3.1.4 and 3.1.5 are also taken from Berbernes and Wilhelmsen [61. The results contained i n Theorems 3.2.1 and 3.2.2 are due to Bebernes and Wilhelmsen [6] (see also Sedziwy [31). Theorem 3.2.3 m a y be found,in Bebernes and Fraker [g]. Exercise 3.2.1 is due t o Markus and Amundsen [ l ] . Section 3.3 contains the results of Kaplan e t al. [l]. Lemmas 3.4.1-3.4.3 are from Bebernes and Schuur [71 w h i l e Remark 3.4.1 is based on Bebernes and Schuur "71 and Yorke [l]. The other results of Section 3.4 are due t o Bebernes and Kelley [lo]. Section 3.5 Consists o f the results of Sedziwy [l]. For related work on continuous dependence, see Bebernes and Gaines [41, Gaines [1,21, Klaasen [l],and Ingram [ l l ,

196

Chapter 4 FUNCTIONAL ANALYTIC METHODS

4.0

INTRODUCTION

Many diverse problems in the qualitative theory of d i f f e r e n t i a l equations are concerned with the existence of a solution which belongs t o a specified subset of a given Banach space.

These problems can be treated i n a unified s e t t i n g by

techniques commonly used i n functional analysis.

These tech-

niques suggest themselves i n a n a t u r a l way, when the considerations involving the i n t e g r a l equation obtained by the method o f variation of parameters are translated into a suitable

abstract setting.

This i s the underlying theme of t h i s im-

portant chapter. A variety of nonlinear functional analytic techniques

l i k e the Fredholm alternative, the Schauder's fixed p i n t theorem, the method of a p r i o r i estimates, the notion of admissibility of spaces, Leray-Schauder's alternative, and the degree theory are employed t o investigate existence r e s u l t s f o r boundary value problems i n various ways.

Some of t h e r e s u l t s

are concerned with periodic boundary conditions. We s t a t e Schauder's fixed point theorem and the contraction mapping principle i n a generalized normed space and then u t i l i z i n g these r e s u l t s , prove existence and uniqueness of solutions of a system of i n t e g r a l equations.

As an application

of the l a t t e r results, we derive existence and uniqueness theorems f o r various boundary value problems including a generalized Nicoletti problem. We present an existence and uniqueness r e s u l t f o r nonl i n e a r functional equations i n terms of the theory of set-

197

4. FUNCTIONAL ANALYTIC METHODS

valued mappings. As an application o f t h i s r e s u l t , we consider the question of existence of solutions of general l i n e a r problems. We then proceed t o develop a general theory of l i n e a r problems f o r set-valuea differential equations utilizing the fixed point theorems for set-valued mappings.

Finally, we

prove an existence r e s u l t f o r boundary value problems associated with set-valued d i f f e r e n t i a l equations.

4.1

LINEAR mOBLEM FOR LINEAR SYSTEMS

For an n-dimensional vector x nxn

(xl,.

=

r e a l matrix A = { a . . I J l e t

..,xn )

and an

1J

n

IIXII =

c

i=1

n

IXil,

lbll =

c

IbiJI*

i,j=1

Consider any compact i n t e r v a l A = [OJh] and define C(M) = C(M)[A,Rl t o be the space of continuous matrix functions A(t) = {a. . ( t ) } 1J defined on A with norm

IbII,

=

max[lb(t)II: t

E

A].

As before, C = C[A,R] w i l l be the s e t of a l l continuous vector functions x ( t ) with norm IlxII. = m [ l k ( t ) I I :

t

E

A].

L1 be the l i n e a r space of a l l r e a l n-dimensional vector functions defined and integrable i n A w i t h norm Let

Define

L,l. (M)

t o be the l i n e a r space of a l l r e a l n x n matrix

Functions A(t) = {a. . ( t ) } defined and integrable i n A with 1J the norm 198

4.1. LINEAR PROBLEMS FOR LINEAR SYSTEMS

A ( t ) E L1(M),

For any

lbllo = 1F11,.

thus L1(M)

define

Then t h e mapping J: A(t)

isometrically onto t h e subspace

continuous

3

?i(t)

Co(M) of a l l absolutely

matrix functions t h a t vanish a t

n xn

maps

t

= 0.

We w i l l a l s o be using t h e norm on A ( t )

Let

L(C,Rn)

mappings

denote the l i n e a r space of a l l continuous l i n e a r

T of

C

into

Rn

with t h e u s u a l norm

IITII = sup[llm(t)II: x ( t ) E C,

llxll0

5

11

For a given T i n L(C,Rn), !I? w i l l be t h e induced mapping of C(M) i n t o t h e space of r e a l matrices which t o every matrix A(t) of

in

C(M)

assigns t h e matrix obtained by t h e application

T t o every column of

A(t).

Hence

!I?

i s a linear

continuous mapping and T[A(t)c] = [T"A(t)]c f o r any

c

E

R".

The following preliminary lemma w i l l be important i n subsequent discussions. LFMMA 4.1.1.

function w(t)

K = [A(t)

E

For any nonnegative Lebesgue integrable

on A,

L1(M):

the s e t

I!.A(t)ll5 w ( t )

199

almost everywhere i n A1

4. FUNCTIONAL ANALYTIC METHODS

i s compact. Proof:

be any sequence i n For any i n t e r v a l

Letting

k

i s a closed s e t .

We f i r s t show K

such t h a t

K

+s]

[t,t

C

lh-Ao

A,

Now l e t t i n g s

4 0,

Lebesgue p i n t

t

lb, ( t ) 11 5 a ( t )

J(K)

i s r e l a t i v e l y compact i n

i s compact i n

it follows t h a t

continuous. compact i n

J(K)

K.

L1(M).

i s continuous, we need C,(M).

From the definition

i s uniformly bounded and equi-

By Ascoli's theorem, the s e t C(M).

a t every

and t h i s implies A o ( t ) belongs t o

Since the mapping J-l: Co(M) 4 L1(M) K,

+ m.

k

we have

it i s clear t h a t

It suffices t o show K

of

+ 0 as

we obtain

4 m,

only show t h a t

1'

{%(%)I

Let

J(K)

is relatively

Since a uniform l i m i t of equicontinuous

f'tmctions, which are absolutely continuous, i s also absolutely continuous, it follows t h a t C, (M),

i s r e l a t i v e l y compact i n

J(K)

This completes the proof.

We s h a l l consider the l i n e a r nonhomogeneous system of d i f f e r e n t i a l equations (4.1.1)

X' =

where A(t)

E

L1(M)

and b ( t )

continuous and linear. (4.1.1) (4.1.2)

A(t)x

+ b(t), E

L1.

Let

T: C + R n

be

We s h a l l be interested i n solutions of

satisfying =(t) = r

200

4.1. LINEAR PROBLEMS FOR LINEAR SYSTEMS

f o r a given

r

E

R ~ .

We now s t a t e a well-known r e s u l t exhibiting a r e l a t i o n ship between homogeneous and nonhomogeneous systems. THEOREM 4.1.1. The BVP (4.1.1), (4.1.2) has a unique solution f o r every r E Rn and every b ( t ) E L1 i f and only if the corresponding homogeneous l i n e a r BVP (4.1.3)

X’

(4.1.4)

= A(t)x,

Tx(t) = 0,

has only the t r i v i a l s o l u t i o n

x ( t ) = 0.

Proof: Let U(t) be the fundamental matrix s o l u t i o n of (4.1.3) w i t h U(0) = I. The general s o l u t i o n of (J+.Ll), then satisfies x(t)

=

where (4.1.5)

xo ( t )

=

U(t)

i s a s o l u t i o n of (4.1.1) arbitrary element of

takes the form

R

n

U(t)c + x o ( t ) ,

lt 0

U-’(s)b(s)

ds

such that x o ( 0 ) = 0 and c i s an Thus the boundary condition (4.1.2)

.

T[U(t)c + xo ( t ) ]

=

r

or

Then (4.1.6) has a unique s o l u t i o n f o r any if

201

r

E

Rn

i f and only

4. FUNCTIONAL ANALYTIC METHODS

t h a t is, i f and only i f (4.1.3) and (4.1.4) has only the t r i v i a l solution. Furthermore, when (4.1.6) (4.1.1),

(4.1.2)

(4.1.8)

i s s a t i s f i e d , the solution of

i s uniquely represented by t h e formula

x(t) = u(t)[fi(t)I-l r

- Tko( t )

+ xo ( t ) .

This completes t h e proof of Theorem 4.1.1. We now l e t

S be a s e t i n

L1(M)

s a t i s f y i n g t h e follow-

ing two properties : (i)

IklI, 5 CY

t h e r e e x i s t s a positive constant

such t h a t

CY

f o r each A ( t ) E S,

(ii)

f o r each

A(t)

in

S,

t h e BVP (4.1.3),

(4.1.4) has

only the t r i v i a l solution. From ( i i ) , it follows t h a t (4.1.1),

(4.1.2) has a unique

solution f o r each A ( t ) E S, b ( t ) E L1,

and

r

E

Rn.

I n t h e following discussion, we l e t the mapping T E L(C,Rn) and t h e vector r be fixed and only A ( t ) and respectively. b ( t ) w i l l change i n S and L1, Let Q: S xL1 C be the mapping which assigns t o each pair

(A(t), b ( t ) )

(4.1.2).

in

t h e unique solution of (4.1.1),

SxL1

!t%e continuity properties of

Q w i l l be inferred

from the Cartesian product topology induced i n t h e space L1(M) xL1 in

by t h e norm

11- 1 '

in

L1(M)

and t h e norm

11- Ill

L1,

LEMMA 4.1.2.

If the s e t

( i ) and ( i i ) , then t h e map Proof:

Q

S C L1(M)

s a t i s f i e s conditions

i s continuous.

We can consider the mapping

202

Q

C+,Q

where

4.1. LINEAR PROBLEMS FOR LINEAR SYSTEMS

Here

U(t)

denotes t h e fundamental solution of (4.1.3),

i s given by (4.1.5),

x,(t)

and x ( t )

by (4.1.8).

Let

F: L1(M) -,C(M)

denote t h e mapping which t o each A ( t ) E L1(M) assigns the fundamental matrix of (4.1.3). I n t h e commutative diagram

the mapping

%

i s continuous since from ( i ) it follows t h a t

t h e r e s t r i c t i o n of of

i s continuous.

S

follows immediately from (4.1.5)

Q1

of t h e mapping of

F(S)

F to F(S)

into

The continuity

and from t h e continuity

which t o every

C(M)

assigns i t s inverse matrix U-l(t).

U(t)

in

Finally, t h e mapping

% i s a l s o continuous since from ( i i ) it follows t h a t (4.1.7) holds f o r any U(t) i n F(S) so t h a t t h e matrix [ f i ( t ) ] ’ l i s a continuous fbnction of

U(t).

Hence, we see t h a t

Q

is

continuous thus completing t h e proof of Lemma 4.1.2. IEMMA 4.1.3.

If a compact s e t

sc

L ~ ( M ) satisfies ( i )

and ( i i ) , then there e x i s t s positive constants only on

S

such t h a t

f o r each A ( t ) Proof: -

B, y depending

in

S

and b ( t )

in

L1.

From t h e continuity of t h e r e s t r i c t i o n of the

mapping F t o the s e t follows t h a t t h e s e t

S F(S)

and from the compactness of

i s compact i n C(M).

203

S,

Defining

it

4. FUNCTIONAL ANALYTIC METHODS

(4.1.10)

Ibll,

mex

a =

UEF(S)

b* = c

we deduce easily from (4.1.5)

llxll

5

=

max

II(T%>-~II,

UEF(S)

and (4.1.8)

that

y = ac and

@ = (l+acIITll)ab*.

Thus

i s proven.

Although S xL1

IkJ'lll,

a c l r l + (l+acllTll)ab*lbI1l,

so t h a t , i n (k.l.g), Lemma 4.1.3

m a

UEF(S)

Q

does not i n g e n e r a l map bounded subsets of

i n t o compact subsets o f

f o r c e r t a i n subsets o f

t h i s property does hold

C,

I n p a r t i c u l a r , we have t h e

S xL1.

following lemma.

4.1.4. and ( i i ) and i f

If a compact s e t

D

i s a set i n

(4.1.11)

Ib(t)ll

for

b(t) E D

then

Q(SxD) Proof:

and

t

E

A,

S

C

L1(M)

satisfies (i)

L 1 such t h a t

1. m ( t >

where

m(t)

i s i n t e g r a b l e i n A,

i s a r e l a t i v e l y compact s e t i n

C.

From t h e diagram i n t h e proof o f Lemma 4.1.2 we

have

(4.1.Q) since

Q(S X D ) = %[Q1(F(S) X D ) ] , Q(S

that, for

xD) U(t)

= F(S)

in

xD.

F(S)

From (4.1.5), and b ( t )

in

(4.L.U) D,

x o ( t ) satisfy

llxo llo 5 ab*

m(t) d t

Since

204

= d.

it follows

t h e functions

4.2. LINEAR PROBLEMS FOR NONLINEAR SYSTEMS

for

<s
0

continuous. in

C(M) xC.

that

Q(S xD)

so t h a t t h e functions

Then, t h e set

%

Since

(sF(S) x D )

x, ( t )

a r e equi-

i s r e l a t i v e l y compact

i s continuous, we see, from (4.1.12),

i s a l s o compact, thus completing t h e proof of

Lema 4.1.4. 4.2

LINEAR PROBLEN FOR NONLINEAR SYSTEM3

Using t h e r e s u l t s i n Section 4.1, we prove e x i s t e n c e theorems f o r t h e BVP (4.2.1)

X'

and (4.1.2). A(t,x),

We assume t h a t t h e

and t h e vector f'unction

x f o r almost a l l t in

Rn.

= A(t,x)x

in

+

b(t,x)

nxn

matrix functions

b(t,x)

are continuous i n

A and measurable i n t

for a l l x

Assume, moreover, t h a t t h e functions

are integrable i n

A.

This implies, by t h e well-known

Caratheodory theorem, t h a t s o l u t i o n s of i n i t i a l value problems of (4.2.1) e x i s t l o c a l l y .

For a given l i n e a r mapping

T: C - t R n

and an a r b i t r a r y

r i n Rn, we consider t h e BVP (4.2.1) and (4.1.2). By an a p p l i c a t i o n of t h e Schauder f i x e d p i n t theorem and t h e

vector

r e s u l t s of t h e previous section, we prove t h e following existence theorem.

We use t h e same n o t a t i o n and symbols as i n

t h e previous section.

205

4. FUNCTIONAL ANALYTIC METHODS

TKEOREM 4.2.1. t h a t f o r any

S be a closed set i n L1(M)

Let

A(t)

E

t r i v i a l solution.

t h e BVP (4.1.3),

S,

Assume t h a t

(4.1.4)

A(t,y(t))

S

E

such

has only t h e

f o r each y ( t )

and t h a t

in C (4.2.2)

Then t h e r e exists a t least one s o l u t i o n of t h e BVP (4,2.1), (4.1.2). Proof: y(t)

E

C]

5

The closure

of t h e s e t

i s a closed subset of

g e n e r a l i t y , we may assume

S

5.

S =

N = [A(t,y(t)):

so, without loss of

From Lemma 4.1.1,

S

is

a closed subset o f a compact set and t h e r e f o r e i s i t s e l f W e have f o r

compact.

a

S

= JA a ( t ) at. Let

W: C + R n

t h a t ( i ) is s a t i s f i e d with

be the mapping which t o each y ( t )

in

C

assigns t h e unique s o l u t i o n s of

x'

(4.2.3)

s a t i s f y i n g (4.1.2). show t h a t

= A(t,Y(t))X

To prove Theorem 4.2.1.,

as signs t h e p a i r

P: C + S xL1

which t o each y ( t )

(A(t,y(t)), b ( t , y ( t ) ) )

Indeed, f o r a sequence

{yk(t)} C C

we have by t h e c o n t i n u i t y of

x,

it s u f f i c e s t o

W has a f i x e d p o i n t i n C.

The mapping

to

+ b(t,Y(t))

A(t,x)

that

206

in

C

i s continuous.

such t h a t

and

b(t,x)

with respect

4.2. LINEAR PROBLEMS FOR NONLINEAR SYSTEMS

almost everywhere i n

A

s o by t h e Lebesgue dominated convergence

theorem, we see t h a t

it follows from Lemma 4.1.2 t h a t t h e mapping

Since W = Bp,

i s continuous.

W

Furthermore, since

[y(t) E

Bk=

c:

P

l/yII 5 k ( j

maps

k =

~

J

~

J

O

.

.

SXD,, where Dk [ b ( t ) E L1: Ib(t)II 5 Bk(t), A], we i n f e r from Lemma 4.1.4 t h a t W i s a compact mapping.

into the set

t

E

F i n a l l y , from (4.1.9) we have

f o r any

y(t)

in

Bk.

From (4.2.2),

s u f f i c i e n t l y l a r g e , W maps

Bk

we see t h a t f o r

into itself.

k

Therefore W

s a t i s f i e s Schauder's theorem and has a t least one f i x e d point. This completes t h e proof of Theorem 4.2.1. From Theorem 4.2.1we MEOREM 4.2.2.

derive t h e following theorem.

If t h e r e i s a p o s i t i v e constant

6

such

that

-

J d e t f i ( t ) )> 6

(4.2.4)

i n N e [ A ( t , y ( t ) ) : y ( t ) E C] and i f (4.2.2) holds, t h e BVP (4,2.1), (4.1.2) has a s o l u t i o n . f o r every A ( t )

Proof: -

closed s e t

From Lemma 4.1.1, S =

it follows inmediately t h a t t h e

3 s a t i s f i e s condition ( i ) . 201

From continuous

4. FUNCTIONAL ANALYTIC METHODS

dependence, we obtain t h a t t h e mapping F: A ( t ) + U ( t ) continuous i n in

thus implying (4.2.4) holds f o r any A ( t )

S,

and hence the s e t

S =

is

satisfies (ii).

S

An appli-

cation of t h e proof of Theorem 4.2.1 completes the proof.

4.3

INTERPOLATION PROBLEMS We now apply Theorem 4.2.1 t o t h e well-known N i c o l e t t i

problem f o r nth-order nonlinear d i f f e r e n t i a l equations. For t h e nth-order s c a l a r d i f f e r e n t i a l equation

)...,x ( n - l > ) ,

x(n) = f ( t , x , x (1)

(4.3.1)

we consider the following problem:

tl < t2 < rl,r2,

-**

...,rn'

< tn

in

A and n

given

n

d i s t i n c t points

a r b i t r a r y r e a l numbers

does there e x i s t a solution

x(t)

of (4.3.1)

such t h a t

X(ti) = ri,

(4.3.2) Assume t h a t the s e t

AxRn,

f (t,x)

=

i = l,...,n?

f (t,x1,x2,.

continuous i n

and measurable i n t h a t there e x i s t

t n

f o r each

..,

i s defined i n

xn)

f o r almost a l l t i n A n x E R , Furthermore, assume

x

nonnegative integrable functions

p,(t), ,,. , p n ( t ) and a nonnegative function po ( t , x ) s a t i s f y ing t h e same r e g u l a r i t y conditions as f ( t , x ) such t h a t (4.3.3)

If(t,x)

I 5 p,(t>

lXll

+ P2(t)

14

+ ***

+ Pn(t)lXnI + PO(t,X),

for

k = 1,2,

...,

t h e functions

(4.3.4) are integrable i n

A,

and

(4.3.5) 208

4.3. INTERPOLATION PROBLEMS

we have

r n

1

we arrive at

We can now w r i t e (4.3.1)

i n the form

which i s equivalent t o the following system of d i f f e r e n t i a l

equations (4.3.7)

and

x!I

=

(i

xit1

=

l,...,n-

1),

n

condition (4.3.2) becomes

(4.3.8)

xl(ti)

=

( i = ly...Jn).

ri

The l i n e a r problem (4.3.7),

(4.2.1)J

(4.1.2) w i t h

209

(4.3.8) i s a special case of

4. FUNCTIONAL ANALYTIC METHODS

0

1

o . . .

0

0

1 0 . .

0

0

0 1 0 .

0

0

0

a (tJx) 1

ae(t>x)

A(tJx) =

.

. . . . . . . . . . . .

We e a s i l y see t h a t t h e family S in

L1(M)

0

.

1

.

'

an(tJx)

of a l l matrix functions

of the form

1

o . . . . .

0

0

l o . . . .

0

0

O l O . . .

. . . . . . . . . . . .

0

o o o . . .

1

. . . . . .

0

a2(t)

i s closed i n

4.1.

'

an(t>

L~(M) and s a t i s f i e s condition ( i ) of Section

Condition ( i i ) i s equivalent t o the requirement t h a t the

f'unction

x(t)

0

i s the unique solution o f t h e problem

210

4.3. INTERPOLATION PROBLEMS

n x

( 4.3 .lo)

an(t)x("-l)

=

x(ti)

= 0,

+

**.

+ a,(t>x,

i = l,...,n

w i t h the coefficients

a i ( t ) s a t i s f y i n g (4.3.9). This requirement i s equivalent t o the condition t h a t

x ( t ) = 0 i s the unique function w i t h an absolutely continuous (n 1)st derivative s a t i s f y i n g conditions (4.3.10) almost everywhere i n A, s a t i s f i e s the inequality

- -

(4.3.11)

Ix (n) ( t ) I

5 p n ( t ) Ixn-l(t> I

+ *** +

p,(t) I x ( t ) 1

Finally, from the d e f i n i t i o n of A(t,x) and (4.3.6) it follows that A ( t , x ( t ) ) E S f o r any x ( t ) i n C and that (4.3.5) implies (4.2.2). Thus, by appealing t o Theorem 4.2.1, we have proved t h e following r e s u l t : THEOREM 4.3.1.

For (4.3.1) assume conditions (4.3.3)

-

(4.3.5) hold. If x ( t ) E 0 i s t h e unique solution of inequality (4.3.11) w i t h absolutely continuous (n 1 ) t h derivative s a t i s f y i n g t h e boundary condition i n (4.3.10), then there exists a t least one solution of t h e problem (4.3.1),

- -

(4.3.2). The following uniqueness theorem follows readily.

-< 2

i=1

Assume

x(t)

=0

pi(t)Ix;-Xil.

i s t h e unique solution of

xn(t> = an(t)xn-l(t) 21 1

+

0 . .

+ a,(t>x(t>

4 . FUNCTIONAL ANALYTIC METHODS

t h derivative satisfying (n - 1 )l a i ( t ) I 5 p i ( t ) . Then there e x i s t s a unique s o l u t i o n of the BVP (4.3.1) and (4.3.2).

w i t h absolutely continuous

(4.3.10), whenever

EXERCISE 4.3.1.

Hint:

Prove Theorem 4.3.2.

Show t h a t ( 4 . 3 . E ) implies t h a t conditions (4.3.3)

-

(4.3.5) hold. Then apply Theorem 4 . 3 . 1 t o show the existence of a s o l u t i o n . The uniqueness then f o l l o w s readily. We now employ these last two theorems by giving s u f f i c i e n t

conditions on t h e f'unctions p i ( t ) and the length of t h e i n t e r v a l [a,b] t o ensure t h a t t h e only s o l u t i o n o f (4.3.10) and (4.3.1) i s the t r i v i a l solution. THEOREM 4.3.3.

n

{pi'li=l

where

Assume

with

s a t i s f i e s (4.3.3)

- (4.3.5),

satisfy

n (4 03 13)

f

c k=O

h = b -a.

one s o l u t i o n i n

hk

2%((k- 1)/2)!(k/2)! Then t h e BVP (4.3.1), [a,b

EXERCISE 4.3.2.

I.

pn-k

'

(4.3.2) has a t least

Prove Theorem 4.3.3.

Hint:

Show (4.3.13) implies t h e only s o l u t i o n of (4.3.10) and (4.3.1) i s the t r i v i a l s o l u t i o n . Then use Theorem 4.3.1. We now apply our r e s u l t t o the BVP

(4.3.14)

x"

= f(t,X,X'),

(4.3.15) Consider the system

212

4.4. FURTHER NONLINEAR PROBLEMS

( 4.3 .16)

x" = p 2 ( t ) x '

+

pl(t)x

and assume 2m

=

max lp2(t)/, t~[a,bl

k =

max

t~[a,bl

Ip,(t)l.

Consider t h e following property o f (4.3.16). LFMMA 4.3.1.

If

4m(b - a ) + k(b

(4.3.17)

- a )2 < 2 , TT

then t h e only s o l u t i o n of (4.3.16) s a t i s f y i n g x(t2) = 0

x(t,)

= 0,

is the t r i v i a l solution.

EXERCISE 4.3.3.

Prove Lemma 4.3.1.

F i n a l l y we s t a t e t h e following r e s u l t .

and t h a t (4.3.17) holds.

Then (4.3.14),

(4.3.15) has a t l e a s t

one s o l u t i o n . Proof:

The proof follows immediately from Lemma 4.3.1

and Theorem 4.3.1.

4.4

FURTHER NONLINEAR PROBLEMS We now apply t h e r e s u l t s of Section 4.3 i n a s l i g h t l y

d i f f e r e n t manner.

N a m e l y , we provide conditions o t h e r than

those used i n Section 4 . 1 t o v e r i f y t h e hypotheses of Theorem 4.2.1.

a r e l a t i o n s h i p on a term involving t h e operator p e r t u r b a t i o n term (4.4.1)

we assume

For example, i n s t e a d of condition (4.2.2),

b(t,x). X'

T

and t h e

L e t us f i r s t consider t h e BVP

= A(t)x

213

+ b(t,x),

4. FUNCTIONAL ANALYTIC METHODS

(4.4.2)

Tx

As before, l e t

U(t)

r.

be t h e fundamental matrix o f

(4.4.3)

X' =

and assume t h a t (a)

=

det[h(t)l

+

A(t)x,

0;

t h i s says t h a t t h e r e s t r i c t i o n of t h e operator TA,

T,

c a l l it

t o t h e space of s o l u t i o n s of (4.4.3) i s i n v e r t i b l e .

This

i s equivalent, a s we have seen, t o t h e fact t h a t t h e only

s o l u t i o n of (4.4.3) s a t i s f y i n g

Tx = 0

is the t r i v i a l solution.

We s h a l l a l s o suppose t h a t (b) where

Ik(t)ll

p(t)

5

P(t),

t

E

A,

i s i n t e g r a b l e on A.

Observe t h a t conditions ( a ) and (b) a r e of t h e same nature as conditions ( i i ) and ( i ) i n Sect'ion 4.2, r e s p e c t i v e b .

we

now present an existence theorem i n w h i c h w e u s e t h e v a r i a t i o n o f parameters formula as w e l l as t h e Brouwer f i x e d p o i n t theorem. MEOREM 4.4.1. satisfied. (c)

Assume t h a t conditions ( a ) and (b) are

I n addition, suppose t h a t b ( t ,x)

verifies t h e Carathgodary conditions i n

and t h a t t h e r e e x i s t s , f o r some s u f f i c i e n t l y l a r g e

RXRn

an i n t e g r a b l e fbnction (4.4.4)

b(tJx)ll

i n A,

vk(t),

5

V,(~>J

t

f

such t h a t

AJ

lkll 5 k,

and

Then t h e BVP (4.4.1), each

r

E

(4.4.2)

has a t l e a s t one s o l u t i o n f o r

R ~ . 214

k,

4.4. FURTHER NONLINEAR PROBLEMS

Proof:

We f i r s t prove t h e r e s u l t i n t h e case t h a t

i s independent of

As before, l e t U(t,)

which

t h a t is, (4.4.4)

k,

+

E

R.

0 1, define f o r each

t 5 to t

for

F ( t ) = U(t)c h

>

i

x(t) = c

where

t

for some

For each i n t e g e r

(4.4.6)

holds with

be t h e fundamental matrix of (4.4.3) f o r

U(t)

= I

vk(t) v(t) vk(t).

E

Rn

U-l(s)b(s,x(s - h / i ) ) ds,

U(t)

i s t h e length of

c

A.

The mapping

i s a continuous transformation of For each C +r

r

E Rn,

Rn

C[A,Rn].

t h e mapping

- W(k) J

.t

c

- h / i ) ) ds

U-l(s)b(s,x(s

i s then a continuous transformation of

f i r e o v e r , from ( a ) ,

into

Rn

into i t s e l f .

e x i s t s , i s continuous, and thus t h e

Ti1

mapping

(4.4.7)

- U-l(t)TihJ(t)

c +dl(t)Tilr

x

Lt

U-’(s)b(s,x(s

- (h/i)))

ds

0

i s a continuous transformation o f since

U(t), U - l ( t )

n R

a r e bounded on A,

215

into itself. we have

Moreover,

4. FUNCTIONAL ANALYTIC METHODS

llU-l(t)Tilr

t

- U-l(t)T;h(t)

U-l(s)b ( s , x ( s

Rn

hence t h e transformation (4.4.7) maps M

in

- (h/i)))

ds

i n t o a bounded set

R”.

By t h e Brouwer f i x e d p o i n t theorem t h e r e exists a

c E Rn

such t h a t (4.4.8)

-dl(t)T;h(t)

c = U-l(t)Tilr

U-l(s)b(s,x(s

xj:

- (h/i)))

ds.

0

we can f i n d ccii E M s a t i s f y i n g i such t h a t a subsequence of { c ], which we again c a l l

Therefore, f o r each (4.4.8)

{ c’],

i,

converges. Now with t h e

ci

xi defined by (4.4.6), (4.4.9)

chosen, we consider t h e corresponding that is,

Xi@) = U(t)ci + U ( t )

t

for

E

d1(s)b(s,xi(s

- (h/i))

A,

and Tx.1 = r.

(4.4.10) Furthermore, s i n c e U(t)ci = T i l r

- T’ib(t)

t

U-’(s)b

216

s,xi(s

- (h/i))

ds,

ds,

4.4. FURTHER NONLINEAR PROBLEMS

we have

Ilkt

U(t)U-l(s)b(s,xi(s

-

( h / i ) ) ) dslJ

0

-< Hence,

{xi(t)]

t o see t h a t

//%’r//+ (1f / / C I T / /

{xi(t)]

which we again c a l l

From (4.4.9) and (4.4.10),

i s a s o l u t i o n of t h e BVP (4.4.1),

{x.(t)} 1

Since

llIll = n,

sup

t, s

it follows t h a t

and from (b) we see

217

F i r s t we

Immediately we have

kJ(t)U’l(s)ll.

it follows by l e t t i n g

converges

(4.4.2).

From t h e Gronwall i n e q u a l i t y , we observe t h a t

Using (4.4.11),

v(s) ds-

it i s c l e a r t h a t

We now prove t h e r e s u l t f o r t h e o r i g i n a l case.

o b t a i n a bound on

A

a r e equicontinuous and by Ascoli’s theorem

{xi(t)]

t o a f b c t i o n y(t).

/b-J(t)U-’(s)//$

%reover, it i s easy

a r e uniformly bounded.

a subsequence of y(t)

SUP

t, s

i

+m

4. FUNCTIONAL ANALYTIC METHODS

Now assume the existence o f a k and vk s a t i s f y i n g Consider the vector defined by (4.4.4) and (4.4.5).

s a t i s f i e s Carathgodary conditions as w e l l

Obviously F ( t , x ) as

lb(tJx)ll 5 vk(t) for a l l x

E

Rn

-

and t

xf

=

E

A.

However, we have shown the problem

A(t)z + T(t,F),

-=

Tx

r

has a s o l u t i o n s a t i s f y i n g (4.4.12) w i t h v replaced by vk. Because o f (4.4.l-l) and (4.4.5), we have 1511 5 k, and thus b ( t , x ) = b ( t , x ) . Consequently, x ( t ) i s a s o l u t i o n o f the BVP (4.4.1), (4.4.2). This completes the proof of Theorem 4.4.1. C O R O L M Y 4.4.1.

Assume conditions (a) and (b).

Suppose

that

( c ' ) b ( t J x ) s a t i s f i e s Carath6odary conditions and f o r each p > 0 , t h e r e e x i s t s an integrable function v ( t ) such P

that

(4.4.13 )

llb(t,x>II 1. v p ( t >

and

218

4.4. FURTHER NONLINEAR PROBLEMS

Then t h e BVP (4.4.1), Proof:

(4.4.2) has a s o l u t i o n .

From (4.4.14),

it follows t h a t f o r

E

>

0

such

that

Now l e t

k

be chosen s o t h a t it a l s o

satisfies

Hence,

I n e q u a l i t i e s (4.4.16) and (4.4.17) imply t h a t inequalities (4.4.4) and (4.4.5) hold w i t h v k ( t ) replaced by vI(t) + Ek. Thus we may apply Theorem 4.4.1 t o assert the s t a t e d result.

219

4. FUNCTIONAL ANALYTIC METHODS

Reduce t h e conditions so t h a t t h e second h a l f of

Hint:

t h e proof of Theorem 4.4.1 i s applicable. strutting a new function

-b ( t , x )

Do t h i s by con-

which i s bounded by an

i n t e g r a b l e function by using conditions (i)and ( i i ) . t h e s o l u t i o n i s bounded by an appropriate number where

and b

Show

-b

agree.

def

TA

E

fi=

0

U&2)

0

0

0

0

0

0

-

. . . . . .

We now look a t a more g e n e r a l r e p r e s e n t a t i o n f o r t h e operator

T.

I n p a r t i c u l a r , l e t us consider t h e S t i e l t z e s

integral

220

4.4. FURTHER NONLINEAR PROBLEMS

(4.4.18)

F = F(t)

where

t

E

A.

dF

fi

i s an

X

= 1"J

nxn

Then f o r any s o l u t i o n

X E

C[A,Rn],

matrix of bounded v a r i a t i o n f o r u

of (4.4.3)

r

T = T u = T U c = A JA dF uc u A

and thus hypothesis ( a ) i s equivalent t o (4.4.19)

det L d F U # O ;

D = /A dF U has an inverse . 'iD A Hence we immediately have t h e following extension of

t h a t i s , t h e matrix Theorem 4.4.1. THEOREM 4.4.2.

Assume (4.4.19) holds as w e l l a s conditions

(b) and ( c ) of Theorem 4.4.1.

Then t h e BVP (4.4.1),

(4.4.18)

has a s o l u t i o n . REMARK 4.4.1.

I n t h e preceding r e s u l t s , namely Theorems

4.4.1 and 4.4.2,

observe t h a t we have assumed t h a t t h e range

of t h e o p e r a t o r

T

i n (4.4.2) l i e s i n

necessary as t h e range of

m.

Rn.

This i s not

T may be a subset o f

f o r any

Rm

The proofs a r e e x a c t l y t h e same where, i n s t e a d of t h e

condition Rn + R m

det

fi #

0,

i s one-to-one.

we assume t h e operator I n fact, f o r t h e case

s o l u t i o n o f t h e BVP (4.4.1),

G(t,s)

fl =

3

fi:

Rn,

the

(4.4.18) i s equivalent t o finding

a s o l u t i o n o f t h e i n t e g r a l equation

where

TA

i s t h e Green's matrix defined as

221

4. FUNCTIONAL ANALYTIC METHODS

tO+h

(-U(t)Dil

dF U(t)U-'(s)

+ U(t)U-l(s),

t < s,

Of course, a fixed point theorem could be applied d i r e c t l y t o

However, t h e proof of "heorem 4.4.1

t h i s i n t e g r a l equation.

i s quite f l e x i b l e i n t h a t it was not necessary t o assume m=n. We again consider the questions posed i n Sections 4 . 1 - 4.3, t h a t i s , finding solutions of the BVP (4.4.20)

X'

= A(t,x)x

+ b(t,x)

s a t i s f y i n g (4.4.18). L e t us now assume t h a t instead of (a),

(a')

Ib(t,x)ll

5

p ( t ) , t E A, x E Rn,

A(t,x) where

satisfies p(t)

is

integrable on A. Let

in

ti < ti+y be any i n f i n i t e sequence o f points

Eti3;=l,

A.

Consider the s e r i e s

(4.4.21)

R(t,s,x) = I

+ o t rt-

+ [If A(t,x)

U(t)U-'(s)].

j sJ

A(tl,x)A(t2,s)

dtl dt2 + * * *

S

i s independent of

x,

then

R(t,s,x) =

It i s easy t o v e r i w t h a t t h i s s e r i e s converges

absolutely and uniformly with respect t o

(t,s,x),

t,s E A

i s a continuous

and

f'unction of

x

E

Rn

and thus

R(t,s,x)

for

(t,s,x).

We have the following r e s u l t . WORE!M

A(t,x)

4.4.3.

Consider the BVP (4.4.20),

s a t i s f i e s ( a ' ) and b ( t , x ) 222

satisfies

(4.4.18) where

.

4.4. FURTHER NONLINEAR PROBLEMS

(c") v(t)

[/b(t,x)(I 5 v ( t )

i s i n t e g r a b l e on

for a l l

idet

id XEC [A,Rn]

Then t h e BVP (4.4.20),

E

AxRn,

where

A.

Moreover, assume t h a t (4.4.22)

(t,x)

Ja

dF R ( t , t O , x ) I

=

> 0.

(4.4.18) has a s o l u t i o n .

Since t h e proof i s similar t o t h a t of Theorem

Proof:

4.4.1 we s h a l l only sketch it. For a f i x e d i n t e g e r x e C[A,R*]

i,

c define

and f o r each

as

x ( t ) = c,

t ,< to, P t

we have

Using (4.4.18),

which can be w r i t t e n i n t h e form B(c)c = b ( c ) . The matrix

B(c)

t h e f'unctions

x,

vector -xandare.

b(c)

a r e continuous i n

Now condition (4.4.22) implies t h a t

B(c)

c

since

i s bounded and

By an a p p l i c a t i o n of t h e Brouwer f i x e d point

has an inverse.

theorem we can o b t a i n f o r each i n t e g e r ci which l i e i n a bounded set f o r a l l

a set of s o l u t i o n s

i, i.

Using t h e same technique as i n Theorem 4.4.1 we have t h e existence of a

y

E

C[A,Rn]

and

223

c E Rn

such t h a t

4. FUNCTIONAL ANALYTIC METHODS

n t

n

It i s not d i f f i c u l t t o show that

y

(4.4.20),

i s a s o l u t i o n of the BVP

(4.4.18). We have mentioned t h a t one application o f Theorem 4.4.1 i s the N i c o l e t t i BVP. We now show an application of Theorem 4.4.3. EXAMPLE 4.4.1.

Consider the equation

x(k) where

A

=

i s a parameter.

we derive (4.4.20)

where

,...,xk-1),

Acp(t,X,X 1 Letting

n = k

224

+

1. I n t h i s case we obtain

4 5 . GENERALIZED SPACES

We want the matrix F then t o obey the condition

s,

f

R(t,to,x)

that is, the only polynomial P ( t )

0,

of degree less than o r

equal t o k satisfying the condition

=o

i s p(t) e 0.

4.5

GENERALIZED SPACES

x,y

If

E

Rn,

we s a y

x

5y

if and only if xi L y i )

i = l,...,n.

DEFINITION 4.5.1. generalized

norm

Let E be a r e a l vector space. A for E is a mapping I)* : ,1 E - t R n denoted

bY I/xIIo = (al(x),

.

0

,an ( 4 )

such that

(a)

llxIIG 3 0,

(b)

IIXI/~=

(c)

llMllG = Ihl IIxllG, that is,

(a)

I~X+Y

ai(x) = O

+

0

that i s ,

2

cri(x)

i f and only i f

x = 0,

f o r a l l i i f and only i f

bll,

+

Ibll,,

0

for a l l i; that is,

x = 0; cri(W) = Ihlcui(x);

that is,

ai(x+y)

(.yi(Y>.

For each x E E,

and

E E

225

Fin,

E

> 0,

let

5

4. FUNCTIONAL ANALYTIC METHODS

BE(x) = [y E E: Ib-xll, < &I. Then [BE(x): x E > 0 1 i s a b a s i s f o r a topology on E.

REMARK 4.5.1.

f

E,

E E

Rn,

It i s not d i f f i c u l t t o see t h a t every

generalized normed space

(E,

I*I,)

2

has an equivalent

I~xI/~

and For example i n R , = ( lxll, 1x2 1 ) IlxII = m a x ( ( x l ( , (x2I) are equivalent. For purely algebraic (ordinary) norm.

and topological considerations, it i s immaterial whether we view

a s a generalized norm space o r an ordinary norm space.

E

Such concepts as convexity, closure, completeness and compactness remain the same.

We do, however, have more f l e x i b i l i t y working

with generalized spaces.

We s h a l l need the following terminology. An A-matrix i s a nonnegative matrix

DEIFIXITION 4.5.2. S

such t h a t

i s positive d e f i n i t e .

I-S

A p o s i t i v e d e f i n i t e matrix

such t h a t

x*Sx > 0

S w i l l be any

f o r a l l x E Rn.

n x n matrix

We w i l l use the following

properties of a positive d e f i n i t e matrix S: (i) (ii)

det S

>

0,

a l l the principalminors of

S

a r e positive

definite, (iii)

i f a l l the off-diagonal elements of

positive then (iv)

(I

a r e non-

S-l i s nonnegative,

if

i f f o r some

S

S

> ~~

m, I - S

- s)-l = C”n=0

sn.

EXERCISE 4.5.1.

then

0,

m

ciz0Sn

converges i f and only

i s positive d e f i n i t e i n which case

Prove ( i )

- (iv).

We now s t a t e the Schauder fixed point theorem and cont r a c t i o n mapping theorem i n a generalized normed space. mOREM

4.5.1.

Let

E be a generalized M a c h space and 226

4 5 . GENERALIZED SPACES

let

F

C

E be closed and convex.

continuous, then

i s completely

has a f i x e d point.

T

I n view o f Remark 4.5.1,

Proof:

T: F + F

If

we ma;y view

E

as an

ordinary Banach space with an equivalent ordinary norm. Theorem 4.5.1.

Then

becomes t h e c l a s s i c a l Schauder-Schonoff theorem,

DEFINITION 4.5.3.

Let

generalized metric f o r

E

be a r e a l vector space. A n d: E X E --f R such

i s a mapping

E

that (b)

d(xJy) = d ( y ? x ) ; d ( x J y ) 2 0 and d(x,y) = 0

(c)

d(x,z)

elements of

5

d(x,y)

+

i f and on*

d ( y J z ) J where

x = y;

if

x, y, z

a r e any

E.

THEOREM 4.5.2. space and l e t

Let

T: E + E

E

be a complete generalized metric

such t h a t

d(=?S) 5 Sd(xJy), where

S

i s a nonnegative matrix such t h a t f o r some m, Sm

i s an A-matrix.

Then

F’urther f o r any

x E E

T

has a unique f i x e d p o i n t

x*.

and

We leave t h e proof a s an exercise. EXERCISE 4.5.2.

Hint:

Prove Theorem 4.5.2.

Use ( i v ) t o show T?x 227

i s a Cauchy sequence.

4. FUNCTIONAL ANALYTIC METHODS

Then use the same arguments as i n the c l a s s i c a l case. COROLLARY 4.5.1.

space and l e t

T: E

Let 4

E

E such t h a t

d(%Ty)

where S such t h a t point

x*

be a complete generalized metric

_< Sd(X,Y),

i s a nonnegative matrix.

If there i s an xo E E Snd(Txo,xo) converges, then T has a fixed such t h a t

xiZo

x*

= lim F X o . n,tm

4.6

INTEGRAL EQUATIONS Let

J = [a,b]

be a fixed i n t e r v a l and l e t

be a vector such t h a t

M.

..,pn)

We w i l l consider the space

[ J ] i s the space of Lebesgue pi-integrable functions. pi i s a generalized Banach space with the generalized norm

where

L P

1 5 pi 5

p = (pl,.

L

b

To avoid confusion, i f f ( t ) = Ja g ( t , s ) ds and f E L b P' Ilfllp. We wish t o consider a then we write 111, g ( t , s ) dsllp vector i n t e g r a l equation (4.6.1) where

F [ x ] ( ~ )= K: J X J x R n + R n

b

J'a

K(t,s,x(s)) ds

and b: J -+R

operator defined by

228

n

.

+ b(t), Let

K be the

4.6. INTEGRAL EQUATIONS

nb - K(t,s,x(s))

K[x](t) =

a

We w i l l denote t h e i t h coordinate of

K(t,s,x)

...,Kn(t,s,x)).

K(t,s,x) = (%(t,s,x),

hence

ds. by

Ki(t,s,x);

A similar

notation w i l l be employed f o r other vector-valued functions. We s h a l l need t h e following assumptions: (HI) that is

K. ( t , s , x ) 1

K.(t,s,x)

s a t i s f i e s t h e Carathe'odory conditions; i s continuous i n

1

and measurable i n

(t,s)

x

f o r each fixed

f o r each fixed

x.

(5) //K(t,s,x)llG5 M ( t , s ) / / x / l G + r ( t , s ) ,

M: J

XJ

(t,s)

where

i s a nonnegative measurable matrix-valued r: J X J 3 Rn i s a nonnegative measurable vector-

Rn2

function and

valued function.

.

( 5 ) f o r some

p = (pl,. . , p ) t h e operator M defined n by M[x](t) = ./,[M(t,s)x(s) + r ( t , s ) ] ds maps L into L b P P' where S M(t,s)x(s) ds$ 5 S//x$,, x E L (H4) P' i s a nonnegative matrix such t h a t f o r some m, Sm i s m Ab

I/. ,

matrix

.

(Hg) ( i )

IIK(t,s,x)

(ii) LEMMA 4.6.1.

t h e operator

If

K maps

EXERCISE 4.6.1.

Hint: -

- K(t,s,Y)IIG 5 M(t,s)IIx

IIK(t,s,O)IIG

5

-

s a t i s f i e s (H1) (%), then

K(t,s,x) L P

into

L P'

Prove Lemma 4.6.1.

i s a simple function and show

F i r s t assume x(s)

t h a t t h e a s s e r t i o n holds.

Then assume

measurable function and show t h a t Let

x E L

P

-YIIG,

r(t,s).

and using (H2), (H )

3

x(s)

i s an a r b i t r a r y

K(t,s,x(s)) conclude t h a t

i s measurable. Ki(t,s,x(s)) E

L (J2). An immediate application of f i b i n i ' s theorem yields pi the r e s u l t . 229

4. FUNCTIONAL ANALYTIC METHODS

THEOREM 4.6.1.

(i)

b(t)

E

-

L

Assume t h a t (H1) - (H4) are s a t i s f i e d and

P’ t h e operator

-+ L i s completely continuous. P P Further any fixed point x Then F has a fixed point i n L P of F i n L satisfies P

(ii)

K: L

.

We w i l l apply Theorem 4.5.1 t o obtain o u r r e s u l t .

Proof:

+ b(t),

F: L 3 L i s completely P P continuous from ( i i ) ; it i s s u f f i c i e n t t o show t h a t there

Since F [ x ] ( t ) = K[x](t)

e x i s t s a closed convex s e t which i s invariant under

-

F.

Now we have from (H2) (H4)

where

11 =

Section 4.5, ‘J

111,b r ( t , s )

-1 = (I-S) q.

ds$

+

(/b(/p. By property (iv) i n

i s invertible and

I-S

Then

S‘J

+

11

=

B = [X E L

u,

.

P’

B

Thus

4B

F: B

and, by Theorem

Let

Define

llxll,

i s closed and convex.

Obviously

(I-S)-’ > 0.

5 ‘J1. For

x

E

B,

we have

4.5.1, F has a t l e a s t one

fixed point i n B. hreover, if

x

i s any fixed point of

that 230

F,

we obtain

which implies

and since

-

(I S)"

2 0, we conclude t h a t

( 5 )-

Assume t h a t (H1), ( € I 5 ) are s a t i s f i e d and t h a t b ( t ) E L Then F has a unique fixed point x* E L P' P' Further for any x E L we have P' THEOREM

4.6.2.

x*(t) = lim Fn[xl(t) n-)m

and

23 1

4. FUNCTIONAL ANALYTIC METHODS

where

S

>0 -

and f o r some m, Sm i s an A-matrix.

Theorem 4.5.2,

Hence by

F has a unique f i x e d p o i n t which m a y be

obtained by successive approximations. Theorem 4.5.2 implies (4.6.2)

The estimate i n

holds thus concluding t h e proof

of Theorem 4.6.2.

4.7

APPLICATION TO EXISTENCE AND UNIQUENESS

In t h i s s e c t i o n we wish t o consider t h e BW x'

(4.7 1)

= f(t,x)

n

, By a s o l u t i o n of (4.7.1) and (4.7.2), we mean an a b s o l u t e l y continuous m c t i o n which s a t i s f i e s f : JxRn + R

where

(4.7.1) 0

almost everywhere and passes through t h e p o i n t s

(ti+

If we l e t

y = x-x

0

0

0

x = (x1,x2, 0

transforms (4.7.1)

...

0

), then t h e s u b s t i t u t i o n n and (4.7.2) i n t o ,X

Solving (4.7.3) and (4.7.4)

i s equivalent t o f i n d i n g a

f i x e d point of t h e i n t e g r a l operator

(4.7.5)

Gi[YI(t) =

gi(S,Y(S))

232

G[yl(t)

given by

i = l ~ * * * ~ n *

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

LEMMA 4.7.1.

The following i d e n t i t y holds.

i n which 1, -1, 0,

ti 5 s 5 t, t < s < ti, otherwise.

Hence (4.7.6) holds. We now s t a t e our main existence r e s u l t , a s p e c i a l case o f which i s t h e N i c o l e t t i BVP considered i n Section 4.3.

THEOREM 4.7.1.

Assume t h a t

233

f(t,x)

satisfies the

4. FUNCTIONAL ANALYTIC METHODS

Carath6odory conditions and t h a t

(ii)

t h e r e i s a matrix

Sm i s an A-matrix and

Then t h e BVP (4.7.1), Moreover any s o l u t i o n x*

n q E R,

where

(4.7.2) has a s o l u t i o n x* E L P satisfies t h e estimate

lIx*-x O IIP

(4.7.8)

Proof:

such t h a t f o r some m,

S = (u. .) =J

5

.

(I-s)-lq,

and

We first observe t h a t

continuous function of

t

;1

i

a

ij

(s)"

ds

is a

so t h a t t h e left-hand s i d e of (4.7.7)

i s defined. We w i l l apply Theorem 4.6.1 t o show t h a t t h e operator

G

given by

where in

L

P' (4.7.1),

K(t,s,y)

i s defined i n Lemma 4.7.1,

has a f i x e d point

By our previous remarks t h i s w i l l imply t h a t t h e

(4.7.2)

has a s o l u t i o n . 234

BVP

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

We now show hypotheses (H1) and ( i i ) of Theorem 4.6.1. V e r i f i c a t i o n of (H1).

- (H4) hold

as w e l l as (i)

We have

K i ( t J s > y ( s ) ) = q i ( t J s ) g i ( s J Y ( s ) ) = $ i ( t J s ) f i ( sJY(s) + Since

fi

xo)*

s a t i s f i e s t h e CarathGodory conditions so does

gi(t,y) = fi(t, y+xo). be measurable,

Hence

Moreover

qi(tJs)

Ki(t,s,y)

i s e a s i l y seen t o

s a t i s f i e s t h e Carathgodory

conditions, verifying (H1). V e r i f i c a t i o n of (H2).

We observe t h a t

n

n

Thus

where

and

ri(tjS) =

n

C

M. . ( t , S )

j=1

Hence

(%)

IX.0

J

‘J

I

+ Iqi(tJs) Ici(s).

i s verified.

V e r i f i c a t i o n of (%).

Let

235

y E L

P

,p =

(plJ ...,pn);

4. FUNCTIONAL ANALYTIC METHODS

then

n

Applying Tonelli's theorem t o t h e positive and negative p a r t s

/,b

En M. . ( t , s ) y . ( s ) j = 1 ij

of (4.7.9), we see t h a t is a measurable function of

J

t.

we have

where

a

ij

Ni =

(s)

E

Since

Ll[a,b].

a

ij

1

ds

(s) E Lqj[a,b], q > 1,

3-

Therefore our estimates y i e l d

cYz1 llaijllqj 1bj"p. +

constant depending only on conclude

1b71

a i j ( s ) ds Since

F'

n

c

ds

Furthermore,

+ ci(s)

from Holder's inequality.

+ri(t,s)

Mij(t,s)y.(s) J

+ ri(t,s)

236

ci E L , we Pi

is a

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

Thus

(3) holds. Verification of (Hk).

in (ii).

Let

We wish t o show

that is,

For a fixed

i,

let

Then

237

S = (a ) ij

be the matrix given

4. FUNCTIONAL ANALYTIC METHODS

n

(by Holder s i n e q u a l i t y )

n

Hence (H4) i s s a t i s f i e d . A l l t h a t remains i s t o v e r i f y conditions ( i ) and ( i i ) of Theorem 4.6.1. We observe immediately t h a t b ( t ) F: L since P 0. Hence ( i ) holds. i n t h e case under consideration b ( t )

V e r i f i c a t i o n of ( i i ) .

L +L P

P

We show t h a t t h e operator

G:

defined by

i s completely continuous.

To prove t h i s we w i l l use Helley's

f i r s t theorem which we s t a t e here.

HELLFX'S FIRST THEOREM.

Let

bounded family of fhnctions on 238

3 be a n i n f i n i t e uniformly

[a,b]

whose members a r e

4.7, APPLICATION TO EXISTENCE AND UNIQUENESS

uniformly bounded in variation; that is, there is an M > 0 b [ f1 is the such that for each f E 5, v [f1 1. M, where a a total variation of f on [a,b]. Then there is an infinite

'?

sequence in 5 which converges pointwise on

[a,b].

Now let

B where (cy1,a

=

[Y E LP'* IIY/IpI a],

is an arbitrary element in RY, that is, cy cy ), cyi 2 0, i = 1, n. Then for each y n

cy

*,...,

...,

= E

B

and hence

2

Mi,

where Mi only depends on a.

239

Thus for each y

E

B, Ki[y]

4. FUNCTIONAL ANALYTIC METHODS

are uniformly bounded in variation on [a,bl. Since K.[y](ti) = 0 for each i, we conclude that IKi[y](t)l 1

5 Mi;

that is, K.[y] are uniformly bounded for y E B. 1 Let y (t) be any sequence in B and apply Helly's first

n

theorem to select a subsequence, again denoted by y,(t), such that K[yn](t) converges pointwise on [a,bl to some function z(t). Since IKi[yn](t)( 5 Mi, we have by the Lebesgue dominated convergence theorem that IIK[yn] Thus K: L + L

P

P

- zllp + 0

as n + m.

is completely continuous and hence (ii) is

satisfied. Hence we apply Theorem 4.6.1 to obtain a solution y(t) of (4.7.3), (4.7.4). n u s x(t) = xo + y(t) is a solution of

(4.7.1), (4.7.2). We also have from Theorem 4.6.1 that

where

This completes the proof of Theorem 4.7.1.

REMARK 4.7.1.

Our proof of verifying (ii) essentially

handles the case when p i < m for all i. If all pi> 1, then all p i < m and we may apply Ascoli's theorem to verify

(ii). Then the general case in which 1 5 pi 5 m follows by combining the cases 1 5 pi C m and 1 C p i s m. We now apply Theorem 4.6.2 to obtain a unique solution of

(4.7.1), (4.7.2).

240

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

THEOREM 4.7.2.

Assume t h a t

satisfies t h e

f(t,x)

Carathhodory conditions and t h a t

(l/Pi)

+

(ii)

t h e r e exists a matrix

where f o r some m,

s"

= 1;

S = (u. .) 1J

i s an A-matrix.

(4.7.2) has a unique s o l u t i o n x*

2

0

such t h a t

Then t h e BVP (4.7.1),

L Moreover, x* P' be r e a l i z e d by successive approximations and satisfies E

can

where

Proof:

We wish t o apply Theorem 4.6.2 t o t h e operator

G

given by

Since much of t h e argument i s t h e same as in Theorem 4.7.1 we w i l l only sketch t h e proof.

As i n Theorem 4.7.1, K ( t , s , y ) s a t i s f i e s (H1). Moreover (H3) and (H ) follow a l s o as i n Theorem 4.7.1. 4 V e r i f i c a t i o n of (H ). We f i r s t show ( i ) of (H?) holds.

5

24 1

4. FUNCTIONAL ANALYTIC METHODS

Observe t h a t

we immediately have t h a t (ii) of ( H ) i s s a t i s f i e d with

5

r i ( t , s > = Iti(t,s)fi(s,xo)I Observe t h a t (H2) follows from (H ), (H1),

5

Theorem 4.6.2 may be applied t o t h e operator

a unique s o l u t i o n p

of (4.7.3) and (4.7.4).

and ( i ) . G

Thus

t o guarantee Hence x * ( t ) =

x o ( t ) + y*(t) i s t h e unique s o l u t i o n of (4.7.1), (4.7.2) and x* E L Moreover x* can be obtained by successive apP proximations. Finally, by l e t t i n g yo = 0 we have from

.

Theorem 4.6.2 t h a t

which implies

This completes t h e proof of Theorem 4.7.2. I n both Theorems 4.7.1 and 4.7.2 we observe t h a t i f 242

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

i s continuous i n both

f(t,x) C(l)

t

and

then we o b t a i n a

x,

s o l u t i o n of t h e BVP. We now study t h e BVP

(4.7.10) (4.7.11) where

+ f(t,X,X')

XI1

.(a)

f : [a,b]

=

0,

= x(b) = 0,

xRz + R .

Using t h e Green's function we o b t a i n some i n t e r e s t i n g estimates on t h e s o l u t i o n s of (4.7.10),

(4.7.11) by applying

the previous techniques of t h i s s e c t i o n . R e c a l l t h a t t h e Green's function xtt = 0, x ( a ) = x(b) = 0

G(t,s)

associated with

is

(s

- a)(b - t)/h,

s

5 t,

(t

- a ) (b - s)/h,

s

2

G(t,x) = where

h

=

b-a.

t,

Then a s o l u t i o n x ( t ) of

(4.7.12)

XI1

+

f(t) = 0

s a t i s e i n g (4.7.11) has t h e form (4.7.13)

G ( t , s ) f ( s ) ds.

We now o b t a i n an existence r e s u l t f o r t h e BVP (4.7.10), (4.7.11) a s w e l l a s estimates on t h e

L1 norm and

of t h e s o l u t i o n s ( i n Chapter 1 we discussed t h e THEOREM 4.7.3. cy

and

@

Let

L,

f ( t , x , y ) E C[[a,b] x R 2 , R ]

be nonnegative numbers such t h a t (cyh2/4)

+ Bh < 1.

243

L2 norm norm). and l e t

4. FUNCTIONAL ANALYTIC METHODS

If(tJxJy)

I

_< ~ 1 x 1+

BIYI

r(t>J

+

r ( t ) E L1[aJb], then (4.7.10), (4.7.11) has a s o l u t i o n . Further every s o l u t i o n s a t i s f i e s

where

(4.7.14) and

(4.7.15) (b)

If

(t JxlJY1) - (t J x2JY2) I 5 a I x1- x2 1

I

+

@ lY1- Y2 IJ

(4.7.11) has a unique s o l u t i o n satisfying (4.7.14) and (4.7.15) w i t h r ( t ) = f(t,O,O).

then (4.7.10),

Prove Theorem 4.7.3 by using Theorem

EXERCISE 4.7.1.

4.6.1 i n (a) and Theorem 4.6.2 i n (b). Here our generalized space E i s the space of % flxnctions x ( t ) = (x,(t),x,(t)) w i t h norm IIxII1 = ( IlxllI1~ Ih2111). A similar theorem i s now presented f o r the (.12 norm. THEOREM 4.7.4.

f ( t J x J y ) be continuous and l e t a and p be nonnegative constants such t h a t Let

(ah

2 2 /TI- ) + (@h/.rr) < 1.

(a> If I f ( t J x J ~ ) l5 ~ 1 x 1+ where

r(t)

E

BIYI

L2[aJb], then (4.7.10),

+

r(t>J

(4.7.11) has a solution.

Every solution s a t i s f i e s moreover,

(4.7.16)

IIX

12

5

h2 2

T ~ - C -p-rrh Y ~

244

Ilr

12

4.7. APPLICATION TO EXISTENCE AND UNIQUENESS

and (4 7.17 1

then (4.7.10),

(4.7.11) has a unique s o l u t i o n which s a t i s f i e s (4.7.16) and (4.7.17) w i t h r ( t ) = f(t,O,O). EXERCISE 4.7.2.

Prove Theorem 4.7.4.

We now wish t o consider t h e BVP

( 4.7.18)

X"

+ A2x .(a)

(4.7.19)

=

f(t,X,X'),

=

x(b)

= 0,

where f : [a,b] xR2n - t R n and A i s a constant matrix of order n. Using the techniques of Chapter 1, we consider the associated nonhomogeneous l i n e a r problem (4.7.20)

X"

2

+A x

= f(t)

and express t h e solution of (4.7.18),

(4.7.19) i n terms of the

Green's function f o r the problem 2

( 4.7.21)

X" + A X = 0

and (4.7.19).

I n order t o guarantee t h e uniqueness o f solutions of

(4.7.18) and (4.7.19), and hence the existence of a Green's fwnction, we assume the eigenvalues o f A, j = lJ**-Jn have the property t h a t

SJ

(PI

hj

#

kTr/(b-a),

k = O,It:1,+2,

This allows us t o deduce t h a t s i n A(b 245

- a)

... . i s i n v e r t i b l e and

4. FUNCTIONAL ANALYTIC METHODS

we may obtain, a f t e r some computation, t h a t the Green's function is

G(t,s) =

{

A-'(sin

Ah)"

s i n A(b

-t)

s i n A(s

- a),

s

A-'(sin

Ah)-1 s i n A(b

- s)

s i n A(t

- a),

s >t.

(4.7.20), (4.7.19) i s

Hence, the unique solution of x(t) = further, x'(t)

s,"

=J

b

a

5 t,

G ( t , s ) f ( s ) ds;

G t ( t , s ) f ( s ) ds.

To obtain an existence theorem f o r t h e BVP (4;7.18), (4.7.19) we need t h e following preliminary computations. If B = (bij) i s a matrix of order IBI

5

ICI,

n,

then l e t

IB

I

=

(

then

Ic W

sin BI =

2k+l

2k+l

k=0

1.

a,

- k=0

Similarly, lcos

BI

5

(e

1'1 + e-ICI)/z.

We now present an existence theorem. 246

lbij

I ).

If

4.1. APPLICATION TO EXISTENCE AND UNIQUENESS

THEOREM 4.7.5.

Let

f ( t J x J y ) be continuous and assume

that + r ( t ) , where P and (a) IIf(t,X,Y)IIm _< PllxII, + Qb//, n a r e nonnegative constant matrices and r E C[[a,b],R+].

Assume t h a t t h e eigenvalues of

be a nonnegative

A

have property (P).

2 n x 2 n matrix i n which

Moreover, assume f o r some m, Sm i s an A-matrix. BVP (4.7.18),

solution

(4.7.19) has a s o l u t i o n .

x(t)

R e c a l l here

r(t)

=

f(t,O,O). Prove "heorem 4.7.5.

A solution x ( t )

satisfies

rb

247

Then t h e

Furthermore any

o f (4.7.18) and (4.7.19)

EXERCISE 4.7.3. Hint: -

Let

satisfies

Q

4. FUNCTIONAL ANALYTIC METHODS

Apply Theorem 4.6.1 where now the operator i s defined as

F = (F1,F2)

and

Show t h e hypotheses of Theorem 4.6.1 a r e s a t i s f i e d i n a manner similar t o t h a t used t o prove Theorem

4.6.1. The only new

point i s t h e v e r i f i c a t i o n of (H ). I n t h i s case show

4

and

This then implies

IM(t,s)l

-<

(l/h) S

and thus

The conclusion of the theorem follows readily.

4.8

METHOD OF A PRIOR1 ESTIMATES Here we s h a l l present theorems on the existence of

solutions of boundary value problems, with both l i n e a r and nonlinear boundary conditions, using t h e method of "a p r i o r i estimates. "

This technique can be described as follows. 248

4.8. METHOD OF A PRIOR1 ESTIMATES

Suppose t h a t it i s required t o prove t h e existence of a solution, where t h e solution i s chosen from some topological space

i n other words, it i s required t o show t h a t a

H;

c e r t a i n subset

H

G

of t h e problem).

i s nonempty (the subset of solutions

To do t h i s , we construct a sequence of

problems of a s i m i l a r type, t h a t i s , a sequence of s e t s

..,

G _C H (p = 1,2,. ) approximating the o r i g i n a l problem P i n t h e sense t h a t g E G (pk --f CQ), g + g implies t h a t k' k ' k ' > 0, h g E G. We f'urther construct a s c a l a r function w(h) -

E

which w i l l give an estimate such t h a t t h e condition EG w(gp) P P subsequence g

""*g

<

w i l l imply t h e existence of a convergent

CQ

+g. Finally, it i s shown t h a t each of t h e *k satisfyapproximating problems has a t l e a s t one solution g P ing sup w(gp) < CQ. This implies t h a t G i s nonempty. P Suppose i n some l i n e a r topological space H we have a

set

G

5H

and a mapping

L of t h e space

H

into

Rn.

We

a r e i n t e r e s t e d i n showing t h e existence of a solution of t h e problem (4.8.1) where

a

E

i s given.

Rn

value problems,

I n applying t h e method t o boundary

i s t h e s e t of solutions of t h e given

G

Assume t h a t there e x i s t s a s c a l a r

d i f f e r e n t i a l equation.

lower semicontinuous function w

in

H

s a t i s f y i n g the in-

equalities 0

(4.8.2)

-< w(h) -<

w(ah)

w(h1 + h2)

CQ,

5 w(h)

whenever

,< w(hl) 0

+ w(hZ),

< a < 1.

Suppose t h a t t h e following conditions a r e s a t i s f i e d : (i)

The operator

continuous operators

L

can be w r i t t e n as t h e sum of two

LO + L ~ ;

249

H,

4. FUNCTIONAL ANALYTIC METHODS

(ii)

there e x i s t s a s e t

0 G

0

(4.8.3)

v

depends continuously on

0

0

f o r any

a E Rn

and t h e solution

a;

there e x i s t s a "comparison problem"

(iii)

*

0 Y E G ,

where

f o r which the problem

L v = a

V E G ,

has a unique solution

-C H

Ly=b,

i s a continuous operator mapping

L*

H

into

R"

e x i s t s and i s unique f o r any b E yb X C Rn, we have supybEGow(yb) < m.

whose solution f o r any sphere

8

and

The preceding conditions guarantee the existence of an operator

A

defined as

= L*V a

Aa

which i s a continuous, one-to-one mapping of and i s hence a homeomorphism.

where

B

(iv) that

E

Rn

into i t s e l f

Thus

.

-1 A

The s e t

g E G k' k' and f o r any b

G

and E

Rn

can be approximated by s e t s

g + g as pk + m implies t h a t k ' each of t h e problems

st

P

g E G;

L*s = b

s E G P' has a unique solution

such

G

depending continuously on b

n = sup w(s,p -Yp) < PER b cRn

250

m.

and

4.8. METHOD OF A PRIOR1 ESTIMATES

(v) The condition sup w(gp) < m implies the existence P of a convergent subsequence g + g. P ( v i ) For any M > 0 and b y a E Rn, there e x i s t s a sphere

such t h a t f o r any b E X, s E G, w(s

X C Rn

- yb) < M,

we have 0

A ( L yb THEOREM

- LS + a )

X.

E

I f conditions ( i )

4.8.1.

- (vi)

are s a t i s f i e d ,

then the problem (4.8.1) has a t l e a s t one solution.

sphere X

M = 1+

Let

Proof:

and f o r the given

according t o ( v i ) .

of the space

Define mappings

i n t o i t s e l f by the formula

Rn

a,

choose the

T P

(p = 1,2)

0 b T b = A ( L y -Ls + a ) P b p Xp = T X.

and s e t

same center as f o r any for

p

an

E

k

+ w

E

and

X,,

> 0,

E

with a radius

X

> 1,

> p,. >1

Let

P

be a sphere having the times greater.

E

that there exists a

p

E

such t h a t

Suppose the claim i s f a l s e . and a sequence

%=T

*k

k

#

b

XE.

b

k

E X

Set

-< sup w(s;-yb) p,b

by condition ( i v ) . a subsequence

sk = s , bk +

Xp

- XE

C

Then t h e r e e x i s t s

such t h a t

W(Sk)5 s'p(w(Sk-Y k bk

We claim,

w(Y

+bo k then

bk

b

as

1)

+ sup W(Yb) < b EX

w

Then ( i v ) and (v) imply the existence of

s of ki

s k' 'ki

25 1

+s

E G.

Hence

4. FUNCTIONAL ANALYTIC METHODS

since t h e lower semicontinuity of w(s - y

implies t h a t

'ki ) < inf w(sb - y b ) < M. bo - i ki ki

However, then, f o r i

w

s u f f i c i e n t l y large,

I

%.

for a l l

E XE

1

-> I.

This i s a contradiction and t h e claim i s proved.

Hence Xp C -X

for

E

p

s u f f i c i e n t l y large.

we have by t h e Brouwer fixed point theorem

T X + X p E XE, P: t h a t f o r each p

there exist

b E X P

Tb = b PP P hbreover,

E

P

--f

Since

0

p + M.

as

A ( L 0yb

P

+ &

and

E

P

such t h a t

P'

Hence, t h e equality

Lsb P + a) = b + E P P P

implies t h a t 0

LSp

bP

= a + L y -B(b bP P =E+Bb

Since

P

sup w(sEp) < P

+ E )

P

we obtain from (v) the existence of a

of

sp

Isbpi }

and thus

Pi P i s a solution of (4.8.1).

REMARK 4.8.1. Go

b

Pi sb + g , Lsbpi + a

such t h a t

If the operator

Pi Lo

1

basis f o r

G

,

Pi

i s linear, the s e t 2 , , vn i s some

i s l i n e a r and n-dimensional, and v , v 0

p+m.

M,

subsequence g

as

=B(b + & ) + a P P

...

then the statement t h a t the problem (4.8.3)

i s uniquely solvable and t h a t t h e solution depends continuously

252

4.8. METHOD OF A PRIOR1 ESTIMATES

on

a

i s equivalent t o the statement t h a t the determinant

A

=

0 1 0 2 On det(L v , L v ,...,I, v )

is nonzero.

We apply now the r e s u l t s t o t h e boundary value problem

$(x

l J . . . Jm )~=

a k

(k = l J . . . , n J

where

fi

a l l ni

>

where

n = n

1

+

.- -

(nm-U a r e defined f o r t E [ t1' t 2 1, -m < x 1, ,xm < 0, and the f'unctionals are defined on the s e t

...

4

of functions

derivatives

x ( t ) = { x i ( t ) ] J tl 5 t 5 t2, f o r which a l l the (ni- 1) xi are absolutely continuous. We assume

s a t i s f i the Carath6odoqy conditions; t h a t is, f i s (nm4 measurable i n t f o r fixed xlJ...,x , continuous with m (nm4 respect t o (xlJ...,x ) f o r each fixed t, and f o r any m M > 0 there e x i s t s an integrable fhnction M ' On (tl't2) such t h a t Ifi/ 5 f M ( t ) f o r a l l i and t E (tlJt2), and that

IIx II

fi

< MJ We m a y write problem (4.8.4) i n the form

(4.8.5) Here

Dx = 0, H

Lx

=

a.

i s the space of functions

x ( t ) = {xi(t)], (ni-1) f o r which a l l the derivatives x are

tl 5 t 5 t 2 J absolutely continuous and the norm on H i s

253

i

coJ

4. FUNCTIONAL ANALYTIC METHODS

n -1

m IIX

Let

max

=

1

be the s e t of solutions of

G

The operator then

It

i

Dx = 0,

where

i s obtained from D by s e t t i n g a l l f

Do

i s the s e t of a l l solutions o f t h e equation

Go

0

i

= 0,

Dx=O. Let t h e comparison problem be the Cauchy problem 0

Dy=O

We thus have t h e following r e s u l t . Suppose t h a t t h e following conditions are

THEOREM 4.8.2.

s a t i s f ied: (i) xi,

Let

...,xm( n m 4,

0 L = L

(ii)

+

L

5

where

fo(t)

where

Lo

t

for

E

L

and

L1

a; (iv) a sphere X

For any M C

Rn

>0

[tl,t2];

a r e continuous operators i n

The boundary value problem

a unique solution f o r any

and all

can be w r i t t e n i n t h e form

H; (iii)

[tl,t2]

i s integrable on

fo

The operator

,

1

Ifi\

0

0

D v = 0, L v = a

has

a E R ~ , depending continuously on and f o r any

such t h a t for any

254

a E Rn,

b E X, s E G,

there e x i s t s and

4.8. METHOD OF A PRIOR1 ESTIMATES

w(s

- yb)

< Mj we have 0

A ( L yb

-b

+ a)

E

x

(see t h e n o t a t i o n i n Theorem 4.8.1). Then t h e BVP (4.8.5) has a t l e a s t one s o l u t i o n . Proof:

We use Theorem 4.8.1 t o prove t h e r e s u l t ; it

thus s u f f i c e s t o v e r i f y conditions ( i v ) and (v) of Theorem

4.8.1, s i n c e conditions ( i ) - ( i i i ) , ( v i ) follow immediately, Observe t h a t each

f

i

can be approximated by

is Lipschitz and s a t i s f i e s an estimate of t h e form Ifi

P

I

_< f o ( t ) , where

fo(t)

i s i n t e g r a b l e on

f

i P

[tlJt2].

which Then

conditions ( i v ) and (v) follow from t h e Lipschitzian p r o p e r t i e s

, t h e d e f i n i t i o n of w(x), and t h e continuous dependence P of d i f f e r e n t i a l equations on i n i t i a l d a t a . This concludes t h e of

fi

S i m i l a r l y t h e following r e s u l t s may be

proof o f Theorem 4.8.2. obtained. THEOREM 4.8.3. 4.8.2 hold. If Lo Q

> 0,

p = constant

-

Suppose conditions ( i ) ( i i i ) of Theorem i s p o s i t i v e l y homogeneous (Loax = Q PLx,

> 0)

and f o r any

lim lla-PL1axll = 0, a400 sP ' w (x) < M a t l e a s t one s o l u t i o n . EXERCISE 4.8.1.

THEOREM 4.8.4.

M

> 0,

we have

then t h e BVP (4.8.5) has

Prove Theorem 4.8.3. Suppose t h a t conditions (i) and ( i i ) of

i s l i n e a r , A # 0, -1 1 and f o r any M > 0 we have limwm S U ~ ~ ( )la~ )L ax11 < ~= 0 , then t h e BVP (4.8.5) has a t l e a s t one s o l u t i o n . Theorem 4.8.2 hold.

EXERCISE 4.8.2.

If t h e operator

Lo

Prove Theorem 4.8.4. 255

4. FUNCTIONAL ANALYTIC METHODS

REMARK 4.8.2.

Observe t h a t i f the boundary operator L 0 1 i s linear, then we may assume L = L and L = 0. Then Theorem 4.8.4 essentially says t h a t the BVP (4.8.5) has a 0 0 solution i f the problem D x I 0, L x = 0 has only the t r i v i a l solution. This r e s u l t has been proved using other techniques i n Section 4.3. 4.9

BOUNDS FOR SOLUTIONS W ADMISSIBU SUBSPACES

I n t h i s section our treatment of the previously considered BVP

(4.9- 1)

X' =

(4.902)

A(t)x + f ( t , x ) , LX = r,

is quite different from that of e a r l i e r sections.

I n particular,

t h e treatment r e s t s on a characterization of the class of linear

d i f f e r e n t i a l equations (4.9.3)

X'

= A(t)x

+ b(t)

f o r which there e x i s t s a solution satisfying (4.9.2). Let J be any compact i n t e r v a l and we mqf think of L as a continuous linear mapping from C[J] onto Rn. Denote by V the inverse image of any r E Rn. W e see t h a t V is a closed l i n e a r variety o f codimension m i n C. W e first examine under what conditions (4.9.3) has a t least one solution belonging t o V whenever b E C = C[J,Rn], Corresponding t o some fixed t o E J, define cp: C - + C t o be the mapping which associates w i t h each b E C the particular solution of (4.9.3) that equals 0 a t to; that is,

256

4.9. BOUNDS FOR SOLUTIONS IN ADMlSSIBLE SUBSPACES

where U(t)

i s a f'undamental matrix of

x' = A(t)x.

be the mapping whose value a t each

Y: Rn + C

solution of

x ' = A(t)x which equals

s at

s

E

Let i s the

Rn

to; t h a t i s

Jr(s) = u(t)v-~(t,)s.

Thus every solution x

of (4.9.3) has the unique representation

x = +(s)

s = x(t,).

+

cp(b)

where

(4.9.3) has a solution i n V

i f and only i f the equation

L o $(s) = r

(4.994)

- L o q(b)

has a solution i n Rn. n L e t Y = L o Jr(R ), CJ = - L O cp B

Then

Hence, it follows t h a t

and define

= CJ -1( - r + Y ) .

is e i t h e r empty o r a closed l i n e a r variety i n

B

the l a t t e r case, every equation (4.9.3) with b l e a s t one solution i n V,

E

B

C.

In

has a t

that is, satisfying (4.9.2).

Xo of L O Jr i s a closed X1 be the complement of Xo and l e t P be the projection of Rn onto Xo. b r e o v e r , the r e s t r i c t i o n of L o $ t o X1 i s an isomorphism of X onto 1 Observe that the n u l l space

linear subspace of

Let

I n addition, there i s a constant

Y. z

Rn.

E

Y,

there i s a unique

and llx,ll 5 hllzII. lemma. IJ?,MMA 4.9.1.

x1 E X1

> 0,

such t h a t

i n which f o r each L o

Jr(xl) = z

We essentially have proved the following

If

B

i s nonempty, there exist positive

constants a, p, y such t h a t given any xo E xo, (4.9.3) has f o r every b E B a unique solution x E V with PX(tO)=xO for which (4.9.5)

IIxlI

5

allx,

11

+ Bllrll + y b 11.

257

4. FUNCTIONAL ANALYTIC METHODS

From Lemma 4.9.1,

c: (xoJb) + x

induces a mapping continuous.

we see t h a t i f

For, consider

x X B into

V which i s

0

x0 XB,

(XoJbl)J (GOJb2) i n

such t h a t

with b = b 1 -b2

of

- c(z0’ b 2)

the mapping w = c(xo,bl)

i s nonempty, (4.9.3)

B

i s a solution of (4.9.3)

p w ( t ) = xo - x o h

0

This implies from (4.9.5) t h a t

then

and

L(w) = 0.

(4.9.6) As we pointed out before, i f linear variety.

r

then

Y,

E

This occurs i f and only i f

l e a s t one solution belonging t o

For

V.

B

i s a closed

x’ = A(t)x has a t t h i s i s always

r = 0

true.

LEMMA 4.9.2.

Jr i s a continuous mapping of Rn onto Rn, then B i s a topological direct summand of C; t h a t i s C = A @ B, where A and B are closed l i n e a r v a r i e t i e s . Proof: Rn

onto Rn

of

Rn,

If

Since

r

E

and

Y

Lo

P O

i s a continuous linear mapping of

L O cp o Jr

whose n u l l space i s a topological d i r e c t summand

there e x i s t s a continuous l i n e a r injection

which i s the right inverse of Y =

- 9 0 M of

Rn

onto the supplement of

Rn

into

Lo

Define the injection

Jr.

cpo

and l e t

C

Q be the projection of

We claim that

Y.

M: Rn + R n

is

T = Y OQO@

a continuous projection of C f o r which T(B) = 0, t h a t i s -1 ? (0) = B. Clearly T i s continuous and i s a projection because some

0

c

o

Y

E C,

i s the identity mapping.

fireover,

and t h i s i s equivalent t o

i s equivalent t o

it follows t h a t

c

E

o-’(Y).

Since

r

~ ( c =) 0

Q o O(c) = 0, E

Y

implies

which B = o-’(Y),

= B.

This completes the proof. We now consider the d i f f e r e n t i a l equation (4.9.1). .C-’(O)

convenience introduce, f o r every b

t + (t,b(t))

of

J

for

into

JxRn, 258

E

V,

For

the injection

so we may write

f o

g,,

g,,: for

4.9. BOUNDS FOR SOLUTIONS IN ADMISSIBLE SUBSPACES

t - + f ( t , b ( t ) ) of J i n t o Rn. The a, B, y w i l l always be those referred t o i n Lemma

t h e continuous mapping constants

4.9.1. A necessary condition f o r (4.9.1) t o have a s o l u t i o n be-

longing t o

is that

V

be nonempty and we assume t h i s

B

throughout. We now attempt t o o b t a i n s u f f i c i e n t conditions f o r t h e existence of s o l u t i o n s of (4.9.1) belonging t o

V,

that is,

s a t i s f y i n g (4.9.2). Suppose t h e r e i s a closed b a l l X f

0

gb

E

f o r every

B

b

E

in

Then (4.9.1)

X.

V

such t h a t

gives r i s e t o t h e

mapping

of x

into

XoXB

O f course

V.

C(xo,b)

i s t h e unique s o l u t i o n

of t h e l i n e a r d i f f e r e n t i a l equation

(4.9.7)

X'

which belongs t o t i n u i t y of b +f

0

gb

2

of

+ f(t,b(t))

= A(t)x

and s a t i s f i e s

V

P x ( t O ) = xo.

The con-

follows from t h e continuity o f the mapping into

X

and t h e c o n t i n u i t y of

B

u: Xo x B 4 V .

Clearly every u E X f o r which C(x0 , u ) = u f o r some xo i s a s o l u t i o n of (4.9.1) i n V with Fu(tO)= x Thus we 0'

need conditions f o r which t h e mapping V

has a f i x e d point f o r some THEOREM 4.9.1.

such t h a t ball in IlbII

5

r

(4.9.1),

llxll

5

s

cya Xo

+

Let

llrll

a

+ yp 5

with c e n t e r a t

implies

f

0

gb

E

B

xo

-> 0, s

0

s

of

X

into

Xo.

>

0, p

>0

be constants

and denote by X 0 t h e closed and r a d i u s a. If b E V and

and

(4.9.2) has a s o l u t i o n x and

E

b 4C(x0,b)

xo

Px(tO) = xo.

259

[If

gbll

5

p,

f o r each x

then t h e BVP E

Xo

with

4. FUNCTIONAL ANALYTIC METHODS

Proof: subset of

The b a l l X = [b E V: because

C

i s closed.

V

assumptions, we have

C(xo,X)

[ v ( t > : v E ~ ( ,xX I ]

set

over, any

0

v

C

)I -< s ]

By Lemma 4.9.1and our

f o r any

X

i s a convex closed xo E X.

Hence the

i s r e l a t i v e l y compact i n R ~ . m r e -

i s a solution of (4.9.7) f o r some

C(xo,x)

E

Ilb

b E X. and thus s a t i s f i e s

Hence

C(Xo,X)

i s equicontinuous and by Ascoli’s theorem,

r e l a t i v e l y compact i n f o r each

x

E

C(xo,x) = x. (4.9.2).

Hence

such t h a t

This concludes t h e proof.

and denote by and radius

f

(ii)

f

(iii)

H

s.

(i)

0

x EX

i s a solution of t h e BVP (4.9.1),

x

Let

be so chosen t h a t

Ilf.9,

By Schauder’s fixed point theorem,

t h e r e e x i s t a f’unction

Xo,

THEOREM 4.9.2. p

C.

a

-> 0

and positive constant

cya + Bllrll + yp

s

the closed b a l l i n Rn

(9< l), 2ks < p, with center a t 0

Suppose i s Lipschitzian i n

0

5

k, s ,

gb E B

there i s a

f o r every

I xH

for t h e constant

b E V with

bo E V with

IboII5

k;

IlLll 5 S; s such t h a t

I I I P -2b.

Then f o r each xo E Xo there e x i s t a unique solution u of the BVP (4.9.1), (4.9.2) with l/u 11 5 s and X 0 xO i s continuous Pux ( t o ) = xo. b r e o v e r t h e mapping xo -x0 0

i n Xo. Proof: subspace of every

b E X,

The b a l l X = [b E V: 1/b11 5 s ] C.

i s a complete

Ilf 0 gbII 5 p for by Lemma 4.9.1 we can define a sequence {vn]

Since by our assumption

260

4.9. BOUNDS FOR SOLUTIONS IN ADMISSIBLE SUBSPACES

of points i n X

such t h a t

v

0

and v = Z(x v ) for n 0’ n-1 = un(xo), where each

= bo

n > 1. By induction, we may w r i t e

un’* X 0 + X

i s continuous.

f o r any xo

E

foxo

x0.

in E

X

Mediately Clearly ux (4.9.1),

F r o m (4.9.6) we observe t h a t

and t h i s implies that

0’

u

Thus

and xo + u

Xo

vn

X

0

X

0

un

converges uni-

= l i m un(xO) e x i s t s f o r every

i s continuous i n X 0 ’

It follows

and ~ ( x o ~ U)x =0 u 0 f o r every x0 E x0 ’ i s unique. For i f w i s another s o l u t i o n of

uX 0

X

E

0

lbll 5

(4.9.2) s a t i s f y i n g

s

and

Fw(t ) 0

=

x

0’

then

w-u is a s o l u t i o n of t h e l i n e a r equation (4.9.3) with X 0 b = fo%-fogU such t h a t Fv(tO) = 0 and T(v) = 0 . v

=

X

0

Since by Lemma 4.9.1 t h e r e i s only one s o l u t i o n we conclude v

=

c(O,b)

However,

and t h i s implies from (4.9.6)

w = u

X

0

because

ky < 1, thus completing the proof.

When B i s a t o p o l o g i c a l d i r e c t summand o f C we can s a y more. For i n t h i s case, t h e r e e x i s t s a continuous p r o j e c t i o n T of

C

with

Thus, even i f

a-l(O) = B and, f o r every b

f

0

gb

does not belong t o

E

X’ =

A(t)x

-b

f(t,b(t)) -%(t)

261

a unique

B f o r any b

t h e l i n e a r equation (4.9.10)

C,

E

V,

4. FUNCTIONAL ANALYTIC METHODS

has a unique s o l u t i o n x ( t ) x

0

E

xo with

Fx(t ) = x 0

following r e s u l t .

of

s a t i s f y i n g (4.9.2) f o r each This y i e l d s immediately t h e

0’

COROLLARY 4.9.1. Let B be a t o p o l o g i c a l d i r e c t summand C. Let T be a continuous p r o j e c t i o n of C with z-’(O>=B a > 0, s

and l e t

2ypl(BI/ 5 s .

If

> 0,

p

(If gbll 0

>0 5P

be constants such t h a t f o r every

then t h e r e e x i s t s f o r each xo E Xo (Iu((5 s , F%(tO)= xo

(4.9.2) with (4.9.U)

U’

b E V with

a function

0

( ( b (5( s,

satisfying

and

( t ) = A(t)u(t) + f ( t , u ( t ) )

[Here h ( u ) = z ( f

u

(Ya + Bllrll+

- hu(t).

g,)].

EXERCISE 4.9.1.

S t a t e and prove a c o r o l l a r y f o r Theorem

4.9.2 i n a formulation similar t o Corollary 4.9.1. We now apply our previous r e s u l t s t o t h e case i n which J = [O,T]

(4.9.

and (4.9.2) becomes

w

W = ~ ( 0 -)x ( T ) = 0.

Thus we a r e looking f o r a s o l u t i o n of (4.9.1) s a t i s f y i n g t h e p e r i o d i c boundary conditions space o f p e r i o d i c mappings Choose

to = 0

and l e t

x’ = A ( t ) x with and

cp: C + C

U

x ( 0 ) = x(T).

u

C

f o r which

V

is the

u ( 0 ) = u(T).

be t h e fundamental matrix o f

U(0) = I.

Jr: Rn + C

Then t h e mappings

can be w r i t t e n as cp(b): t + U ( t )

+(x): t - + U ( t ) x ,

(4.9.13)

E

Here

JtU - l ( s ) b ( s ) ds,

xo i s t h e n u l l space, and Y t h e range of t h e mapping I U(T) The mapping L cp Jr i s the b i j e c t i o n x -TU(T)x !Thus

-

.

0

0

so t h e assumptions of Lemma 4.9.2 a r e s a t i s f i e d . set

B o f mappings b

E

Hence t h e

C f o r which t h e l i n e a r d i f f e r e n t i a l

262

4.10. LERAYSCHAUDER’S ALTERNATIVE

equation (4.9.3) has at l e a s t one periodic solution i s a topological sununand of

C.

A topological supplement t o

Q

choosing

( i n the proof of m

the projection of of

I - U(T),

of

I-U(T).

projection

and

Y : Rn

Rn

Then -r = Y

onto t h e n u l l space

f o r example, as Yo

of the adjoint

0

B

i s the n u l l space of t h e continuous n CJ: C --f R i s the mapping

Q O CJ where

+ C such t h a t

It follows t h a t Qyb = 0

B

t

--f

(1/T) U(t)U-l(T)x.

i s precisely the s e t of those b

E

C

for

where

1

T

yb = (l/T) U(T)

4.10

a 4.9.2),

which i s an orthogonal supplement t o the range

Y(x):

which

B may be constructed by

U-’(s)b(s)

0

ds.

LFRAY-SCHAUDER’S ALTERNATIVE Let us now investigate t h e general existence problems

based on d i f f e r e n t geometric ideas.

We s h a l l employ, as the

t i t l e of t h e section indicates, the Leray-Schauder‘s theory of topological degree. DEFTNITION 4.10.1. operators defined f o r

ht

To, T1

u E S

P’

with values i n a M a c h space

be completely continuous

where E.

S

P

= [u E E:

llull

5

p],

Then we s h a l l s a y t h a t

are homotopic i f there e x i s t s an operator T(u,h), To, T1 t h a t i s completely continuous on M[0,1] such t h a t T(u,O)

E

TOu, T(u,l) = T1u

for

263

u E S

P

and

T(u,h)

#

u

for

4. FUNCTIONAL ANALYTIC METHODS

IluII = P. We need the following r e s u l t of Leray-Schauder. LFJDfA 4.10.1.

defined on

S

P

Let

be a completely continuous operator

T

E.

with values i n

Suppose t h a t

t o the operator i d e n t i c a l l y equal t o zero. a t least one solution u Ilull

<

x

C(2)

i s homotopic

Then t h e r e e x i s t s

Tu = u such t h a t

P.

Let E

of t h e equation

T

X

E [0,1] and l e t

[ [0, 11,Rn]

(4.10.1) f

where

S(h)

XI1

= hf(t,X,X'),

n n C [ [0,1]X R n X R ,R ] J

E

denote the s e t of functions

satisfying

and t h e general boundary

conditions (4.10.2)

~ ( 0 -A0x'(O) ) = 0,

(4.10.3)

x(l)+A1x'(l) = 0 ,

A.

, A1

being

n Xn

matrices.

Then we can prove t h e following

r e s u l t which we give as an exercise with generous h i n t s .

EXERCISE 4.10.1. i d e n t i c a l l y zero. B

>

0

Ilx(t)ll

5B

Then show t h a t the s e t Hints:

Ao, A1

be positive d e f i n i t e or be

Suppose f u r t h e r t h e r e e x i s t s a constant

such t h a t , i f

(4.10.4)

Let

X

E

[0,1] and

S(X),

we have

t

E

[0,11.

i s nonempty.

Recall t h a t t h e t r i v i a l solution i s the only

solution of t h e homogeneous equation (4.10.2),

E

llx'(t)l( 5 B,

and S(1)

x

(4.10.3)

XI' = 0 which s a t i s f i e s

and t h a t there e x i s t s a Green's matrix

G(t,s) such t h a t (4.10.1) with (4.10.2) (4.10.3) i s equivalent 1 t o x ( t ) = A .fo G ( t , s ) f ( s , x ( s ) , x * ( s ) ) ds. Now use t h e f a c t 264

4.1 1 . APPLICATION OF LERAY-SCHAUDER'S ALTERNATIVE

t h a t t h e map

1 x ( t ) + J 0 G ( t , s ) f ( s , x ( s > , x ' ( s ) ) ds

continuous i n t h e Banach space

c ( ~ [)[O 11 ,Rn] J

i s completely

with the

supremum norm and apply k m a 4.10.1. APPLICATION OF LERAY-SCHAUDER'S ALTERNATIVE

4.11

A s an application of Leray-Schauder

1s

a l t e r n a t i v e , we

s h a l l , i n t h i s section, prove existence of solutions of the d i f f e r e n t i a l system

(4. ll.1) where

XI1

= f(t,X,X')

n n f E C[[O,l]xRnxR ,R

1,

subjected t o the boundary

conditions (4.10.2) and (4.10.3).

Let us begin with the follow-

ing r e s u l t .

THEOREM 4.11.1. (i) definite; (ii)

n

f E c [ [ o , ~X I R ~ X R,R

(t,v)

(4.11.2)

n

V E C(2)[[0,1] xRn,R+],

g E C[[O,l] x R + x R , R - ] , each

Assume t h a t

and f o r V:(tJx)

g(t,u,v)

3

and A ~ A~ , V(t,x)

i s positive d e f i n i t e ,

i s nonincreasing i n

( t , x ) E [O,l]XRnJ

2

g((tJV(tJx)J V'(tJX))J

265

are positive

u for

4. FUNCTIONAL ANALYTIC METHODS

V'(O,x(O)) f o r some

2 cy

(iv)

0, V'(Lx(1))

5

xR+,R]

G E C "0,l.l

such that f o r

u

2

V(O,x(O))

5

aV'(O,x(O))

T E

and t h e r e e x i s t s an

L

>

0

L, t E [O,l],

- (V/Ul2

(l/u) g(t,u,v) and f o r any

and

0

> 0.

(0,1],

1 G(t,v/u)

t h e l e f t maximal solution

r(t,T,O)

of G(t,z),

Z' =

z(T) = 0

r ( t J T J O ) < a0, t E [O,T],

s a t i s f i e s t h e inequality

where

a. = min(+-,l/a);

(v)

the l e f t maximal solution

minimal solution

p(t,O,O)

r(t,l,O)

and t h e r i g h t

of

v' = g(t,2L,v) e x i s t s on

[0,1].

Assume a l s o t h a t f o r each solution x o ( t ) xo(t)

i s defined f o r a l l t E

bounded.

[0,11

or

of ( 4 . l l . l )

either

Ilxo(t)II i s un-

Then there e x i s t s a solution of (4.ll.l) satisfying

(4.10.2) and (4.10.3). Proof: V,

By Lemma

1.14.1 and t h e positive definiteness of

we have f o r each

(4.11.3 )

(h,x)

Ilx(t)ll

We s h a l l show t h a t

< t< 1 Il(p(t)[I and

5 B*,

i.

Since

llv' !lo],

A.

A.

[0,1] xC(')[[0,l],Rn]

= max[/b//OJ

(p E

where

C(')[[O,l],Rn].

we may assume t h a t

vergent t o some all

i s compact i n

and xo

as

hi

i

and

+ m.

lllpll0 =

If not, there

{hi,x.1 ( * ) I with no l i m i t i n

e x i s 5 asequence /Ixi(0)II

-< t -< 1.

0

B*,

S

IlrpI1l

with respect t o norm

s'po

5

E S,

xi(0)

Let

S.

Since are con-

yi = x;(O)

i s assumed t o be positive d e f i n i t e , 266

for

4.1 1 . APPLICATION OF LERAY-SCHAUDER'S ALTERNATlVE

yi = A.-1xi(0),

by (4.10.2),

which converges t o some y

-1

0

= A x 0 0'

From the standard convergence theorem, there i s a solution x o ( t ) of (4.10.1) with I = [O,T)

(a)

C

[0,1]

h = ho

defined on an i n t e r v a l

such t h a t

xo(0) = xo, x ' ( 0 ) = yo

and e i t h e r

I = [O,T]

or

x0 ( t ) cannot be defined continuously on any larger i n t e r v a l as well as a t T; (b) i

+m,

x i ( t ) -+xo(t) on

By (4.11.3),

and x;(t)

-+x;(t)

uniformly, as

I.

we must have

Ilxo(t)II _< B", t E I. Since x o ( t ) i s bounded, the hypothesis of the theorem implies I = [0,1].

t = 1, applying (b) above, it follows t h a t x o ( t ) s a t i s f i e s (4.10.3). Hence (Xo,xo) E S, contradicting our assumption t h a t (A x . ) has no limit point i n S. The s e t S i s therei' 1 fore compact as claimed. We now consider the derivative evaluation f'unction n F: [0,1] XS + R given by F ( t , x ( t ) ) + x ' ( t ) . This function i s continuous and i t s domain i s compact. Hence )$II is bounded by some B1 > 0, which, i n turn, implies t h a t

At

IIx'(t)II I :B1> Let

0

-< t -< 1

and

(h,x)

E

s.

B = m a x [ ~ * , ~ ~ ~n ] . application of Exercise 4.10.1 proves

the s t a t e d r e s u l t . Notice t h a t the assumption A.

i s positive d e f i n i t e i s

crucial i n the proof of Theorem 4.11.1.

i s true, i f

Ao, A1

Whether the r e s u l t

are identically zero, remains an open

question. COROLLARY 4.ll.1.

and f o r some k

2

Assume t h a t

n n

f E C[[O,l] xRnxR ,R

0,

l l ~ 1 1+~ x*f(t,x,y) 2 -k[1 267

+

lkll

+ 1x-y

11

]

4. FUNCTIONAL ANALYTIC METHODS

for

(t,x,y)

E

[0,1] xRnxRn.

positive d e f i n i t e .

Let t h e matrices

Ao, A1

be

is

Suppose t h a t f o r each solution x o ( t )

defined f o r a l l t E [O,l]

or

t h e BIT (4,11.1), (4.10.2),

and (4.10.3) has a solution.

Ilxo(t)I) i s unbounded.

"hen

Next we s h a l l give another proof of Theorem 1.14.1 which depends on bray-Schauder I s a l t e r n a t i v e . Alternate proof of Theorem 1.14.1.

s where

S(h),

= [(h,x): A E [0,13 and

x E S(A)],

as i n Section 4.10, denotes t h e set of functions

x satisfying (4.10.1), is nonempty, since i f and define

Let

(4.10.2), x

E

and (4.10.3).

then

0,

(0,x) E S.

The set

S

(A,x)

Let

E S

m(t) = V ( t , x ( t ) ) ,

so t h a t because of assumption (c), we have (1.14.16) as before. Also since (A,x) E S, 0
2 g(t,V(t,x),V' (t,x))

+ dlhf(t,x,xl)

11

which again leads t o t h e d i f f e r e n t i a l inequality (1.14.18). The r e s t of t h e proof is exactly t h e same as t h a t of t h e proof of meorem 1.14.1 as given i n Section 14 of Chapter 1, u n t i l The s t a t e d conclusion

we a r r i v e a t t h e inequality (1.14.19).

follows, i n t h e present case, by Exercise 4.10.1 and t h i s completes t h e proof.

COROLLARY 4.11.2. hold.

Let hypothesis (a) of Theorem 1.14.1

Assume t h a t f o r some k > 0

Ih.1I2 + x * f ( t , x , y ) -> -k[1 for

d d ( t , x , y ) E [0,1] XR X R

+

.

hII

and

+

m

Ix*Y~]

> 0, + Ullf(t,x,y)II

men t h e r e e x i s t s a solution

f o r the boundary value problem (4.11,1),

268

(4.10.2),

and (4.10.3).

4.12. PERIODIC BOUNDARY CONDITIONS

COROLLARY 4.11.3.

1.14.1 i s s a t i s f i e d .

for

(t,x,y>

E

[0,11

Assume t h a t hypothesis ( a ) of Theorem

For some

k > 0, u > 0, l e t

X R ~ ~ R Then, ~ . if

s a t i s f i e s ~agumo's

f

condition, t h e r e e x i s t s a s o l u t i o n f o r t h e problem (4.11.1), (4.10.2),

and (4.10.3).

We can a l s o give another proof o f Theorem 1.14.2.

Alternate proof of Theorem 1.14.2.

m(t)

Let

S,(h,x)

and

be as i n t h e a l t e r n a t e proof of Theorem 1.14.1.

u

Distinguishing t h e two cases

-> 1

and

0

arguing as i n t h e proof of Theorem 1.14.2,

V h ( t , x ) _> g ( t , V ( t , x ) , V ' ( t , x ) )

 -(N + 1) + llx~(t)ll,

m"(t)

- ( N + 1)

+

~llx~(t)ll

and consequently, as before, we a r r i v e a t 0

-< t -< 1,

(1.14.19)

4.12

using Lemma 1.12.1.

Ilx'(t)((5 y(M), This then implies t h e estimates

and Exercise 4.10.1 then concludes t h e proof.

PERIODIC BOUNDARY CONDITIONS This s e c t i o n provides s u f f i c i e n t conditions f o r t h e

s o l u t i o n s of t h e second-order d i f f e r e n t i a l system

(4.l2.1)

XI'

=

f(t,X,X')

s a t i s f y i n g t h e p e r i o d i c boundary conditions (4.12.2)

~ ( 0 =) x(T),

~ ' ( 0= ) x'(T). 269

4. FUNCTIONAL ANALYTIC METHODS

A solution of the BVP (4.12.1), (4.12.2) will be called a periodic solution. The approach is to establish existence results for the boundary condition

(4.12.3)

x(0) = y = x(T)

and then study the vector field

-

(4.12.4)

U(Y) = X' ( 0 , ~ ) X'(T,Y),

where x(t,y) is the unique solution of (4.12.1), (4.12.3). To solve the BVP (4.12.1), (4.12.2), it is sufficient to prove the existence of a y such that U(y) = 0. We shall assume n n that f E C[[O,Tl XRnXR ,R

1.

LEMMA 4.12.1.

Let Dn be the closed n-ball with radius one, that is, Dn = [x: ((x((5 13. Assume for each y E Dn, (&.12.1), (4.12.3) has a unique solution x(t;y). Further, 0 let there exist a constant N [depending on Dn (interior

-< N Then x(t,y) t E I.

and x'(t,y)

MERCISE 4.12.1.

for 0 5 t 5 T, y

E

Dn.

are continuous in y for each

Prove Lemma 4.12.1 by applying Ascoli's

theorem. Before introducing our main result, we need the following infornation on degree theory. Because the ideas of degree theory are strongly connected with algebraic topology we shall only mention, without proof, the essential theory needed. The idea is to obtain for each f: Sn +Sn an integer (positive, negative, or zero) called its degree. Here S" is the boundary of Dn+l. We shall always assume f is continuous. 270

4.12. PERIODIC BOUNDARY CONDITIONS

For

1

n = 1 t h e degree of

f: S

number of times t h e image point

r o t a t e s around S1 when 1 S In particular, f o r

f(z)

.

z p e r f o m one oriented r o t a t i o n of each

22,.

k = O,+l,

..,

t h e map

+ zk has degree

z

The d e f i n i t i o n of t h e degree of i s a generalization of t h e case details. Let

O:

LEMMA 4.12.2.

then

D(f) = D(g).

property t h a t

x

E

such t h a t and

X

Let

n l 0.

If

Let

n> 0

and

f(-x) = -f(x).

there e x i s t s an x

E

Recall t h a t two O(x,O) = f ( x ) ,

cp(x,*) never vanishes. f,g: Sn + S n f : Sn + S n

Then D(f)

a r e hamotopic, have t h e

i s odd; i n p a r t i c u l a r ,

f ( x ) = 0.

such t h a t

Dn+l

n>1

D(f).

I = [0,1].

X X I +Y

f o r each

for

a r e called homotopic i f t h e r e

f,g: X + Y

e x i s t s a continuous O(x,l) = g(x)

by

f

be two spaces and

continuous maps

f : Sn + Sn

k.

n = 1. We shall omit t h e

Denote t h e degree of X, Y

i s simply t h e

+S1

We now s t a t e our main r e s u l t .

THEOREM 4.12.1.

Let t h e hypotheses of Lemma 4.12.1 hold.

n is a convex homeomorphic image of Do and l e t nbe symmetric about a point z E n. For each y E D define

Assume t h a t

-

U(y) = x ' ( 0 , y ) - x ' ( T , y ) .

an (the boundary of

Let

be t h e continuous mapping of

A

n) onto i t s e l f which maps each y

onto the point which i s symmetric t o Further assume t h a t f o r all y U(y)

and

c

>

0

such t h a t

U(y) = cU(Ay)].

e x i s t s a solution of t h e BVP (4.12.1), Proof:

If t h e r e e x i s t s a

then t h e proof i s complete. field U

does not vanish on

synnnetric about

0,

do not have t h e same d i r e c t i o n [ t h a t is,

U(Ay)

there e x i s t s no

z.

with respect t o

y

an f o r which U(y) f

E

E ail

z,

y

E

Then there

(4.12.2).

an such t h a t U(y)

= 0,

Assume therefore t h a t t h e vector

an.

Since

5

i s convex and of Dn

there exists a homeomorphism g 27 1

4. FUNCTIONAL ANALYTIC METHODS

onto

1

such that

t h a t is, f o r all r E aDn, sn- 1 onto i t s e l f .

i s t h e antipodal map of

g-lAg

Define t h e vector f i e l d

on

cp

Dn

by

cp(r) = U ( g ( r ) ) . By Lemma 4.12.1, continuous.

U

i s continuous, thus implying

vanish on aDn.

an,

does not vanish on

Since U

cp

cp

is

w i l l not

Observe t h a t

cp(r>= U ( g ( r ) ) and

d-r)

= u ( g ( - r ) ) = U(Ag(r)).

From t h e hypothesis on U, we see t h a t w i l l have d i f f e r e n t d i r e c t i o n s on aDn;

#

cp(-r)/M-r)I

cp(r) and that is

cp(-r)

cp(r)/lcp(r>l

Hence t h e vector f i e l d

I - A cp(-r)/lcp(-r)I

$(r, A) = cp(r>/l cp(r>

h 5 1. Since J r ( r , O ) and never vanishes on aDn f o r 0 Jr(r,l) a r e homotopic, t h e i r topological degrees a r e i d e n t i c a l

by Lemma 4.12.2.

$(-r,l)

Since

4.12.3 that t h e degree of t h e degree of exists

-

r

and hence

0

E Dn

Jr(r,O)

= -$(r,l)

we have by Lemma

i s an odd integer.

Jr(r,l)

Hence

i s an odd i n t e g e r which implies t h e r e

such that

U(g(y) ) = 0

Jr(T,o)

where

This implies

= 0.

g(y)

E

a.

)cp;(

= 0

This proves t h e

existence of a s o l u t i o n of t h e BVP (4.12.1),

(4.12.2).

I n our f i r s t application, we s h a l l assume f o r each M > 0,

t h e r e e x i s t s an

N > 0 such t h a t whenever x ( t ) 272

is

4.12. PERIODIC BOUNDARY CONDITIONS

a solution of (4.12.1) defined on

I = [O,T]

Ilx(t)ll

with

5 M,

then Ilx'(t)II ,< N. Further there e x i s t s a constant p 7 N such that i f x ( t ) i s a solution defined on I of the perturbed equation

(4.12.5) with

x"

+

f(t,X,X')

=

Ex,

0

<

,< EO,

&

Ilx(t>ll I M, then IlxT(t>ll_< P. We f u r t h e r adopt the following convention.

are vectors i n R ~ , then l e t AX f(t,X,Y)

- f(t,G3.

THEOREM 4.12.2.

-

x,

-

If x, y, y = x - x, AY = y y, ~f =

-

Let there exist a positive constant R

such t h a t 2

x * f ( t , x , y ) + IIy(( _> 0

(4.12.6) (IxI( = R,

if x-y = 0 ,

and

- n f o r any x, x E R , Ilxll, _< R, x # and y, y E Rn w i t h Ax-Ay = 0. Then there exists a solution x ( t ) of (4.12.1),

lxl

(4.12.2) with Proof:

IIx((5 R. Let

E

>

0

be given with

E

_<

EO.

perturbed equation (4.12.5) and l e t F(t,x,x',E) Then, from (4.12.5), it follows t h a t

(4.12-8)

x * F + llyll

i f . x*y = 0

and

llxll

2

= x-f

2

Consider the f(t,x,x') 2

+ Ellxll + lly112 >_ Ellxll > 0

= R.

Furthermore, from (4.12.7), we observe that

(4.12.9)

AX-AF

+ l l A ~ 1 1=~ AxeAf +

2 &llAXll2 '0 , 273

+

llA~11~

+

Ex.

4. FUNCTIONAL ANALYTIC METHODS

if Axdy

=

0 and

Ax f 0.

We have seen in Section 1.14 that conditions (4.12.8) and (4.12.9) imply that the BVP (4.12.10)

X" =

(4.12.11)

F(tyX,X',E)

~ ( 0 )= z = x(T)

has a unique solution x(t,z,E)

with

IIX(t,z,E>II 5 R,

(4.12.12)

for any z, IIzII 5 R. Let fi = [x: llxll < R]. Then, by assumption there exists a p > 0 such that //x'(t,z,E)// _< p for any z E 5. Thus by Lemma 4.12.1, the vector field U(z,E) = x'(O,Z,E)

-

-x'(T,z,E)

is continuous on n. Again assume that U(z,E) does not vanish for z E ail. Letting r(t) = 3llx(t,z,E)II 2, we find

rl (t)

(4.12.13)

=

x(t,z,E)*x' (t,z,E),

and (4.12.14) From (4.12.8) and (4.12.12)- (4.12.14), it follows that r'(0) < 0 < r'(T) for any z Therefore,

E

an which implies Z.U(Z,E) < o for z

U(ZYE)/llU(Z,E>II

#

u(-z,~>/llu(-z,E)II,

z

E

E ail.

3%

that is, U(z,E) and U(-z,E) cannot have the same direction. By Theorem 4.12.1, we conclude U(z,E) will have a zero in fi for every E, 0 < E < Eo, tfiat is, there exists a solution x(t,E) of the BVP (4.12.10), (4.12.U). By a standard application of Ascoli's theorem, there exists a sequence Ei + O

2 74

4.12. PERIODIC BOUNDARY CONDITIONS

such that x(tyEi) + x ( t ) as i + a and x ( t ) i s a solution of the BVP (4.12.1), (4.12.2). This completes t h e proof of Theorem 4.12.1. We now use t h e theory of d i f f e r e n t i a l i n e q u a l i t i e s d i s cussed i n Chapter 1 together w i t h Theorem 4.12.1 t o derive some r e s u l t s . We s h a l l assume t h a t i n (4.12.1) f i s independent of x ' . I n RL1, consider t h e usual p a r t i a l ordering x 5 y if and only i f xi ,< yi, i = 1, n and x < y if and only i f x . < yi, i = 1,. ,n. Recall t h a t a function ty E C(2)[I,Rn] 1 i s c a l l e d a lower solution of (4.12.1) i f

...,

..

cy"(t) 2 f(t,CY(t)),

(4.12.15)

f3

Similarly,

E

C(2)[I,Rn]

t

i s c a l l e d an upper solution i f

f3"(t) ,< f ( t , B ( t ) ) ,

(4.12.16)

I.

E

Further, we w i l l assume t h a t

t

1.

E

f(t,x)

i s quasimonotone

increasing i n x. THEOREM 4.12.3.

L e t there e x i s t lower and upper solutions

and @ of (4.12.1) with a ( t ) 5 p ( t ) , t quasimonotone increasing i n x on t h e s e t w = { ( t , x ) : u ( t ) ,< x

Moreover, l e t (4-12-17)

cy

Q(0)

5

p(t), t

I

E

cy

E

and l e t

f

be

I}.

and f3 be such that = ~(T)Y

~ ' ( 0 2)

u'(T),

Further, assume t h a t f o r every

B(0)

= B(T),

B'(0)

5

y,

B'(T)*

with

cy(0)

5 y ,< 8(0)

the

BVP

(4.12.18)

X"

=

f(t,x),

and (4.12.3), has a t most one s o l u t i o n x ( t ) such that ( t , x ( t ) ) E w. Then t h e r e e x i s t s a s o l u t i o n of the BVP (4.12.18), (4.12.2). 275

4. FUNCTIONAL ANALYTIC METHODS

[cu,BI = [y: a ( 0 ) 5 y

Let

Proof:

5

Then by

B(0)l.

Theorem 1.ll.1and t h e hypotheses of Theorem 4.12.3 of (4.12.18)

e x i s t s a unique Solution x ( t , y )

there

such that

X(0,Y) = X(T,Y) and

a(t)

5 x(t,y) 5

B(t)

>0 Ilx(t,Y>II, I l x ~ ( t , ~ ) ,< lI N

t h e r e exists an N

for every y

[a,@]. Further [cu,B1 such t h a t

depending on for

Y

E

The vector

[cu,~].

E

field U(Y> = X'(O,Y)-X'(T,Y) i s continuous on (4.12.17),

[cy,@I.

that

fact, if t h e r e exists an Bi(0),

If

U(cu(0)) = 0 i,

cy(0) =

it follows from In

p(O),

and t h e proof i s complete.

15 i 5 n

such t h a t

~ ~ ( =0 )

then it follows again from (4.12.17) t h a t (U(y))i

is

Therefore we need only t o consider zero f o r all y E [a,p]. t h e components (U(y)). f o r those j i n which cu.(O) < f3.(0). J J J W e assume now cu.(O) < B.(O) f o r j = 1 n, the contrary J J s i t u a t i o n w i l l follow by using a similar argument i n a l a v e r

,...,

dimensional s e t t i n g . Thus l e t y

E

an such t h a t

assume U ail

52 =

U(y) = 0,

t h e proof i s done.

does not vanish on 8%

mapping y

If t h e r e exists

[y: a ( 0 ) C y < p(O)]. Let

Otherwise

be the mapping on

A

i n t o t h e point symmetric t o

y

about

b ( O ) + B(O)I/2.

Let that

hi

be a continuous function on

hi(pi(0))

<

0 C hi(ai(0))

and l e t

Consider t h e vector f i e i d defined f o r

276

[cyi(0),Bi(O)] &

y E

>

sd

0

be given.

given by

such

4.12. PERIODIC BOUNDARY CONDITIONS

Thus O(y,E) and O ( A y , E ) do not have the same direction, which implies O(y,E) must have a zero in R. Pick a converging monotonically to zero sequence E1 > E2 > E 3**and let yn be a sequence of O(y,En) in n. This sequence has a convergent subsequence converging to a zero of U(y) in R. This concludes the proof of Theorem 4.12.3. COROLLAHY 4.12.1. Assume that there exists constant vectors Q and p, Q ll 5 Lllx - YII

for some L > 0 where o ,< x, y 5 p. Then the BVP (4.12.1), (4.12.2) has a solution provided L < 8/T2. MERCISE 4.12.2. Prove Corollary 4.12.1.

Hint:

Apply Theorem 4.12.3 by showing there exist at most one solution of (4.12.18) and (4.12.3). This may be done by setting up an integral equation and showing the operator is a contraction mapping with constant T2L/8.

EXAMPLE 4.12.1. Consider the two-dimensional secondorder system 277

4. FUNCTIONAL ANALYTIC METHODS

(4.12.19)

XI'

=

x3- y + p(t),

y" = -x + y3

+ q(t),

a, lq(t)l 5 a. Choose b > 0 so that Then letting f3 = (b,b), o = (-b,-b), we find that f3 is an upper solution and o is a lower solution of (4.12.19). The right-hand side of' (4.l2.19) is quasimonotone increasing in x and y for -h ,< x, y b and is Lipschitzcontinuous in the region. Thus, we may conclude for T sufficiently s m a l l the hy-potheses of Corollary 4.12.1 are satisfied. Thus there exists a solution of (4.12.19), (4.12.2) for T sufficiently s m a l l .

where

Ip(t)l

b3- b - a 2 0.

5

4.13 SET-VALUED MAPPINGS AND FUNCTIONAL EQUATIONS We wish to present here an existence and uniqueness result for nonlinear functional equations, which will be stated in terms of the theory of set-valued mappings. The approach is topological in nature and it permits us to establish the existence of solutions provided a criterion of uniqueness is f'ulfilled. Let E be a Banach space and let n(E) denote the family of all nonempty subsets of E. For a set A in n(E), a mapping H: A --f n(E) is called upper semicontinuous if its graph [ (x,y): y E H(x)I is closed in A X E . The map H is said to be cmpact if, for any bounded subset B of A, the closure of the set UxaBH(x) is compact in E. The map H is called completely continuous if it is upper semicontinuous and cmpact. For a single-valued mapping h: A +E, the upper semicontinuity of the mapping H: x + h(x) is equivalent to the continuity of h, the compactness of H is equivalent to that of h and the complete continuity of H means the complete continuity of h. We now state our fixed point theorem.

278

4.13. SET-VALUED MAPPINGS AND FUNCTIONAL EQUATIONS

space

THEOREM 4.13.1. Let U be a neighborhood of 0 i n t h e E and l e t H: U + n ( E ) be a completely continuous map-

ping such t h a t

x

(4.13.1)

E

H(x), x

E

U

Then, f o r any continuous mapping

h: E + E ,

h(x) - h ( y ) E H ( x - y )

(4.13.2)

x = 0.

implies

for

t h e condition

x-y E U

implies t h a t t h e equation

x

( 4 13 3)

= h(x)

has e x a c t l y one s o l u t i o n . Proof: by u ( x ) = center a t

u

Denote by u ( x )

+x

x

= [y

+ x: y

and of radius

t h e neighborhood of E

u ] and by

K(x)

x

t h e b a l l with

chosen i n such a way t h a t u ( x )

&

contains t h e b a l l with center a t

x

of radius

2 ~ .

Assumption (4.13 .l) implies t h a t t h e mapping where

I

defined

-

T = I h,

denotes t h e i d e n t i t y mapping, i s one-to-one when

r e s t r i c t e d t o t h e neighborhood

U(X),

f o r every x E E,

which,

i n i t s t u r n , shows t h a t (4.13.4) Let

T ( X ~=) T ( x ~ ) , xl S(x)

#

x ~ + K ( x ~n ) K ( x ~ =)

denote t h e boundary of t h e b a l l

claim t h a t t h e r e e x i s t s a

6

> 0,

independent of

$.

K(x).

We

x,

such

that (4.13.5)

Y E s(x)

Suppose t h a t it i s f a l s e .

* h'(y) - T(x) 11 > 6Then t h e r e e x i s t sequences

{x,],

{yn] such t h a t limn+w ( T(Y,) - ~ ( x , ) ) = 0 and Ibn-xnll = for n = l J 2 , . . . S e t t i n g rn = T(yn) -T(xn), we have

.

279

E

4 . FUNCTIONAL ANALYTIC METHODS

and consequently, by (4.13.2), (4.13.6)

Y,

- xn - rn

-

n

H(yn x,),

E

= 1,2,.

Since by assumption t h e closure of t h e set compact, we may suppose t h a t t h e sequence an element

such t h a t

z,

llzI( =

H

is

II 5 E

H(x) { y n - x n - rn} o r , IIX

{yn - xn] i s convergent t o Relation (4.13.6) and t h e

which means t h e same, t h e sequence upper semicontinuity of

u

.. .

E.

imply t h a t

which i s a

z E H(z),

c o n t r a d i c t i o n t o (4.13.1).

it follows t h a t t h e continuous

From t h e condition (4.13.2), mapping every

i s completely continuous.

h x

E

Thus, by (4.13.5),

for

we have

E,

(4.13 7 ) This shows, i n t u r n , t h a t t h e s e t

It i s e a s i l y seen that

T(E) of

#

E,

and l e t Since

T(E).

y

T(E) = E.

y

of radius

i s open,

T(E) 6,

(E,T)

simply connected,

T

Indeed, suppose t h a t

y

does not belong t o

T(x)

T(E).

lying i n t h e neighborhood

we see, by (4.13.7),

that

y

(4.13.7) and t h e f a c t

Relations (4.13.4), t h a t the p a i r

E.

be an a r b i t r a r y element on t h e boundary

On t h e o t h e r hand, f o r a point of

i s open i n

T(E)

i s a covering space f o r

f

T(E) E.

T(E). =

E

Since

imply E

is

i s a homeonorphism, and t h i s completes

t h e proof of t h e theorem.

RFMARK 4.13.1.

It i s easy t o see t h a t i f a mapping

s a t i s f i e s condition (4.13.2), t h e mapping

ha: x

--f

h(x)

- a.

then f o r every

If

b E E

h and

E

E,

so does

t h e mapping

I-h

onto i t s e l f .

i s a mapping of t h e form

A: E + E

E

Therefore, Theorem 4.13.1 may

be s t a t e d i n an equivalent form as follows:

i s a homeomorphism of

a

h

h(x) = A x

+ b,

where

i s a completely continuous l i n e a r operator,

Theorem 4.13.1 y i e l d s t h e e s s e n t i a l p a r t of the f i r s t theorem of 280

4.13. SET-VALUED MAPPINGS AND FUNCTIONAL EQUATIONS

It s u f f i c e s t o s e t

Fredholm.

H(x) = {Ax} and t o observe

t h a t condition (4.13.1) means t h e uniqueness o f s o l u t i o n of t h e

We a l s o note t h a t i n t h e

l i n e a r homogeneous equation

x = Ax.

Fredholm theorem t h e map A

may be noncontractive i n general.

The assumption t h a t t h e map

h

i s continuous, i n Theorem

4.13.1 may be dropped by strengthening t h e conditions imposed on t h e mapping

This i s t h e content of t h e next theorem,

H.

Let

THEOREM 4.13.2.

space

E

and l e t

H:

u

u +n(E)

be a neighborhood of

i n the

0

be a completely continuous

mapping such t h a t t h e implication (4.13.1) holds t r u e and (4.13.8)

H(0) = 0 .

Then f o r any mapping

h: E + E

s a t i s f y i n g (4.13.2),

Eq. (4.13.3)

has e x a c t l y one s o l u t i o n . Proof continuous

It i s enough t o show t h a t

h

i s necessarily

To t h i s end, assume t h a t a sequence

convergent t o

xo.

Then, f o r

n

Exn]

is a

s u f f i c i e n t l y large, we have

( 4-13 9 Suppose now t h a t compactness of converges t o

{h(xn)] H,

yo

#

does not converge t o

t h e r e e x i s t s a subsequence h(xo).

h(xo).

By the

{h(% ) ] n

On passing t o t h e l i m i t i n

which

(4.13.9),

we see t h a t

and t h i s implies, by (4.13,8),

that

yo

- h(xo) =

0.

This

c o n t r a d i c t i o n proves t h e theorem.

We wish t o point out t h a t without t h e assumption (4.13.8), Theorem 4.13.2

i s not t r u e .

For example, i n

281

R2

let

4. FUNCTIONAL ANALYTIC METHODS

H(x) =

[-1,11

for

x = 0,

[-1,$x]

for

x

<

0,

and

3

1

h(x) = q ( x - 7 - [x-$]). It i s easy t o v e r i f y t h a t a l l t h e assumptions of Theorem 4.13.2

a r e s a t i s f i e d except (4.13.8).

The function

not continuous and t h e equation

x = h(x)

h,

however, i s

does not have any

so l u t ion.

4.14

GENERAL LINEAR PROBLEMS A s an a p p l i c a t i o n of t h e t o p o l o g i c a l method developed i n

t h e preceding s e c t i o n , we s h a l l consider t h e question of existence of s o l u t i o n s of g e n e r a l l i n e a r problems.

I n general,

one compares t h e l i n e a r problem under consideration with another homogeneous problem s u i t a b l y chosen and from t h e uniqueness of s o l u t i o n s of t h e second, concludes t h e existence of s o l u t i o n s of t h e f i r s t .

I n t h e present case t h i s comparison

problem involves e i t h e r an equation with set-valued right-hand s i d e or some d i f f e r e n t i a l i n e q u a l i t y . denote t h e family of a l l nonempty, closed, n and convex subsets o f R For A E cf(Rn), we s e t lbll = Let

cf(Rn)

.

sup [Ilpll: p E A]. Let J be a compact i n t e r v a l of n C = C[J,R 1 with t h e norm of uniform convergence,

m a xt c J Ilx(t)II. (1) F

t

E

J xRn

i s a mapping of

IIxII =

into

cf (R").

For each

i s upper semicontinuous with respect t o

and f o r each compact s e t

(4.14.1)

and l e t

Let us assume t h a t

J, F ( t , x )

summable on

R

J,

K C Rn

x, t h e r e e x i s t s a function m(t),

such t h a t

I l ~ ( t , x ) l I1. m(t), 282

t

E J,

x

E

K;

4.14. GENERAL I.INEAR PROBLEMS

(2)

f

i s a mapping o f

JxRn

is continuous r e l a t i v e t o

f(t,x)

is suwnable with respect t o

(3)

L

into

x,

f(t,x)

Rn.

t;

i s a l i n e a r continuous mapping o f

Given t h e mappings

F, f ,

(4.14.2)

and E

t

For each

and f o r f i x e d x

c

into

E:

J,

E

R ~ ,

R ~ .

we consider t h e equation

L,

F(t,x(t))

with t h e homogeneous l i n e a r condition (4.14.3)

Ix= 0

and t h e ordinary d i f f e r e n t i a l equation X'(t) = f ( t , x ( t ) )

(4.14.4)

with t h e l i n e a r condition ~x= r,

(4.14.5)

r c R ~ .

A n absolutely continuous f'unction

a s o l u t i o n of (4.14.2) s a t i s f i e s (4.14.2)

x

E

C w i l l be called i f it

[respectively, of (4.14.4)l

[respectively, (4.14.4) ] almost everywhere

on J.

THEOREM 4.14.1.

Assume t h a t t h e h c t i o n s

-

s a t i s f y conditions (1) (3) and f o r

(4.14.6)

f(t,q)

- f(t,P)

E

Suppose f u r t h e r t h a t t h e h c t i o n of (4.14.2) s a t i s f y i n g (4.14.3).

t

E

F, f ,

L

and

J, p,q E R ~ ,

F ( t , q - PI.

x

I

0

i s t h e unique s o l u t i o n

Then f o r every

e x i s t s one and only s o l u t i o n of (4.14.4)

r

E R",

there

s a t i s f y i n g (4.14.5).

Proof: We wish t o apply Theorem 4.13.1. Consider t h e mapping H of E = C x R " i n t o cf(Rn) such t h a t f o r every point (x,p) its image H(x,p) is a s e t of a l l p a i r s (y,q) given by t h e formula

283

4. FUNCTIONAL ANALYTIC METHODS

y(t) for

u(s)

E

t

= aJ

F(s,x(s))

t h a t f o r every p o i n t

U(S)

ds + p,

q = p

- Lx

and t h e mapping h of E i n t o i t s e l f such (x,p)

i t s image h ( x , p )

is a pair

(y,q)

given by

where

a

i s a f i x e d point i n J.

From (4.14.1),

follows t h a t f o r every compact s e t sup)I\f(t,p)l\:p

E

KI

K

(4.13.2)

problem (4.14.4),

Rn,

i s bounded by a summable fbnction

Ilf(t,O) (I. It i s easy t o see t h a t t h e maps

(4.13.1),

C

H, h

H

it

m(t)

s a t i s f y conditions

and t h a t t h e existence of s o l u t i o n s of (4.14.5)

i s equivalent t o t h e existence of

s o l u t i o n s of t h e f u n c t i o n a l equation (4.13.3). map

(4.14.6),

t h e function

i s compact and t h e map h

Evidently t h e

i s continuous.

Thus t o

complete t h e proof, it i s s u f f i c i e n t t o show t h a t

H

i s upper

semicontinuous. Suppose t h a t

Then we have

and consequently

This, t o g e t h e r with t h e upper semicontinuity of

284

F,

implies

+

4.14. GENERAL LINEAR PROBLEMS

i s t h e distance f'unction. By Lema 4.14.1 (which follows below) it then follows t h a t

where

p(p,A)

~ ' ( t E) F ( t , x ( t ) ) .

(4.14.8)

On passing t o t h e l i m i t i n (4.14.7), (4.14.9)

Y(a> = P,

we obtain, f o r

9 = P

-

t = a,

Ix

and, as a r e s u l t , (4.14.10)

Y(t>

t

=J a

From r e l a t i o n s (4.14.8), that

4.13.1.

Y'(S.1 ds + p. and (4.14.10) it follows

(4.14.9),

and t h e proof i s complete by Theorem

(y,q) E H(x,p)

We now prove Lemma 4.14.1which we used i n t h e preceding proof. LEMMA 4.14.1.

Let

G

be a mapping on

J

into

cf(Rn)

and l e t a sequence {vk} of absolutely continuous f'unctions, n vk: J + R (k = 1,2, ...) s a t i s f y t h e conditions

l h v,(t)

= v(t)

k+m (4.14.11) (4.14.12)

Ivl;(t)I

Proof:

h(t),

t

a.e.

lim p ( v i ( t ) , G ( t ) ) = 0 k+ 00

Then t h e f'unction

( 4.14.13 )

5

for

v

E J

a.e.,

h ( s ) ds <

m,

a.e.

i s absolutely continuous, and

v'(t)

E

G(t)

Define 285

a.e.

i n J.

4. FUNCTIONAL ANALYTIC METHODS

m

( 4.14.14)

m

H(t) = f l conv fl v i ( t ) , i=l

k=i

where conv A denotes t h e smallest convex and closed set containing A. In view of (4.14.12), (4.14.15)

H(t)

C

G ( t ) a.e.

t

E J.

H(t) i s compact, convex, nonempty a.e. measurable i n t and Clearly

( 4.14.16)

IIH(t)ll

5 h(t)

t

a.e.

E

t

E

J,

J.

I n the proof we s h a l l use t h e following property (4.14.17)

( t - s)"

rt

r(H(u), H(s)) du + 0

as

t + s, t f s,

J S

f o r almost every s, where r(A,B) = m&x(p(A,B),p(B,A)), p(A,B) = supxd p(x,B),A,BC Rn. If H(t) reduces t o a (integrable) vector function, t h i s property i s a classical result. For t h e general case, it can be proved by using t h e standard r e a l variable methods. The function v ( t ) i s absolutely continuous because of (4.14.U). Hence the derivative v ' ( t ) e x i s t s a.e. i n J. We s h a l l now show t h a t (4.14.18)

v'(s)

E

if

H(s)

v'(s)

exists

and i f (4.14.17), (4.14.12) f o r t = s are s a t i s f i e d . The inclusion (4.14.18) i s equivalent t o ( 4.14.19)

p

( vO-v(s), t-s

because v t ( s ) exists.

H(s))+O

as

By v i r t u e of (4.14.ll),

p(vi(t),H(t)) and by (4.14.14)

286

I 2h(t)

t + s,

t >

(4.14.16),

8,

4.14. GENERAL LINEAR PROBLEMS

p(vi(t),H(t)) + O

a.e.

i n J.

Hence

Since H ( s ) i s compact and convex, there e x i s t s a unique point %(u) E H ( s ) such t h a t

-

p(vi(u),H(s)) = IIvi(u) Pk(‘>ll>

(4.14*21) the vector

pk(u)

(4.14.22)

gk(t) = ( t - s)-l

We have

(4.14.23)

p(

is integrable and

pk(u) du

w,

H ( s ) ) = k+m lim

It follows from (4.14.22) that

Therefore, we f i n d that

because of (4.14.21),

and consequently 287

p(

E

H(s).

-

vk(t> t-s

4. FUNCTIONAL ANALYTIC METHODS

+ ( t- s ) - l By

v i r t u e of (4.14.20),

Lt

r(H(u),H(s)) ds.

(4.14.23),

(4.14.24),

we obtain

The conand by (4.14.17) we obtain (4.14.19) and (4.14.18). clusion (4.14.13) easily follows because of (4.14.15). The proof i s complete.

Theorem 4.14.1 assumes a particularly simple form, i f Eq.

(4.14.2) reduces t o a d i f f e r e n t i a l i n e q u a l i t y (4.14.25)

+

where w: J xR + R

+

IIX’

II 5 w ( t ,

llxll ),

i s a f’unction satisfying t h e condition

(4) f o r each t E J, w(t,u) i s continuous w i t h respect + t o u and f o r each u E R , w(t,u) i s measurable w i t h respect t o t; the functions sup[w(t,u): u 5 k], k = 1 , 2 are

,...,

suuimable. COROLLARY 4.14.1.

t h e conditions (2) -

and L satisfy (4) and, i n addition, the inequality

Ilf(t,P)

If t h e functions

f, w

- f(tY d I I 5 w ( t , IIP - 911 )

holds and i f the problem (4.14.3),

(4.14.4) has only t h e t r i v i a l s o l u t i o n x = 0, then there exists one and only s o l u t i o n of t h e problem (4.14.4), (4.14.5). For the proof it is enough t o set

F(t,p) = [q: ((qll,< and then apply Theorem 4.14.1. w(t,llpll)] L e t us demonstrate Corollary 4.14.1by an example. 288

4.15. GENERAL RESULTS FOR SET-VALUED MAPPINGS

Consider Eq. (4.14.4) with the boundary condition (4.14.26)

x(a) + k ( b ) = r,

h

> 0,

J

=

[a,bl.

It is easily verified that every solution of the d i f f e r e n t i a l

inequality IIx' 11 < tp(t)llxll where cp(t) i s a positive sumable b function such that I, cp(s) ds < T, satisfying the homogeneous boundary condition x(a) + k ( b ) = 0 i s necessarily t r i v i a l . Indeed, it is w e l l known that f o r an absolutely continuous function x ( t ) satisfying the d i f f e r e n t i a l inequality IIx'I) < cp(t)Ilxll, either x ( t ) p 0 o r x ( t ) i s never 0 on J. Suppose t h a t x ( t ) i s never 0. Then s e t t i n g z ( t ) = x(t)/llx(t)ll, we a r r i v e a t the following contadiction

Thus from Corollary 4.14.1the next corollary follows

immediately. COROUARY 4.14.2.

If the function

f

s a t i s f i e s hypothesis

(2) and the inequality

IIfhP)

- f(t,q>ll 5 cp(t)llp-

911,

where y ( t )

i s positive sumable function verifying I, rp(s) ds < T, then f o r each r E Rn, the problem (4.14.4), (4.14.26) has exactly one solution. b

4.15

GENERAL RESULTS FOR SET-VALUED MAPPINGS

In what follows, we wish t o construct a general theory of l i n e a r problems f o r set-valued d i f f e r e n t i a l equations o r some times called contingent equations, by u t i l i z i n g fixed point theorems f o r set-valued mappings. With t h i s motive, we introduce necessary tools and prove some general r e s u l t s concerning set valued mappings. We s h a l l first s t a t e the Kakutani-Ky Fan fixed point theorem. 289

4. FUNCTIONAL ANALYTIC METHODS

THEOREM

4.15.1. Let L be a locally convex topological

linear space and K a compact convex set in L. Let cf(K) be the family of all closed convex nonempty subsets of K. Then for any upper semicontinuous point to set transformation f from K into cf(K), there exists a point xo E K such that xo E f(xo). We shall now consider some preliminary results. We shall continue to use the same notation of Section 4.14. Let Lp (15 p ,< m) denote the Banach space of all real-valued P will denote its n 3 functions p-summable on J = [a,b]; Ln Cartesian power. In both Lp, LE, the norm w i l l be denoted by

II. Itp*

For a linear normed space E, let, as before, cf(E) denote the set of all closed, convex, nonempty subsets of E. A mapping G: J + cf(Rn) is called measurable if for every p E Rn the distance from p to G(t) is measurable on J. The following lemmas are needed in our main results. LEMMA

4.15.1. If sequences {wi] c Lg, {v,]

C Lp and functions v E Lp and w satisfy the a.e. conditions: Iwi(t)l 5 vi(t) a.e. on J, wi(t) +w(t) on J and llvi- vllp + 0 , then w E LE and llwi-wllp 4 0 .

(1,< p <

m),

The proof is left to the reader as an easy exercise. LEMMA 4.15.2. For a sequence {w,] C L’ and a m c t i o n n L1 satisfying Iwi(t)l 5 cp(t) a.e. on J, there exists k = i,i+l, ...) of a double sequence {A.ik] (i = 1,2 m real nonnegative numbers such that %=1 hik = 1, hik = 0 for sufficiently large k (depending on i), and the sequence

cp E

,...;

m

290

4.15. GENERAL RESULTS FOR SET-VALUED MAPPINGS

converges a.e. on J to a function w Proof: Setting yi(t) = wi(t)/[l

E

1 Ln.

+ cp(t)]",

1

we have

and consequently, we can find a sequence {y 3 and a function 2 Pi yo E Ln such that limi+.. yBi = yo in the sense of weak con2 vergence in Ln. By virtue of the Banach-Saks theorem, one can find a conveniently chosen partial sequence such that

in place of the sequence {y 1. Bi

As the sequences

tend to zero because of (4.1.!j01),we obtain ,,

2

n +n-1

By Riesz's theorem, there exists a sequence of an? n such that CY

lim

n-w

u

n

f o r a l l t E J.

c" i=un

YBi(t) = y,(t>,

an

=

It is now easy to see that

29 1

2

an + an- 1

4. FUNCTIONAL ANALYTIC METHODS

converges f”r a l l values on J

t o t h e function

w =

TI“.

Thus i n order t-o show t h e conclusion it i s 2 s u f f i c i e n t t o s e t Aik = 1/ci if t h e r e e x i s t s an index i

yo/[l +

k = pi,

f o r which

an

LEMMA 4.15.3.

a function y

E

a.e. on J, such t h a t

Lt

of i n t e r s e c t i o n

G(t)

center a t t h e o r i g i n of

to

0

where y(t)

E

It i s enough t o take f o r

Proof:

hi k = 0

Let a map G: J +cf(Rn)

5 rp(t)

IlG(t)II

and s e t

The proof i s complete.

contrary case.

let

5 pi 5 an,

G(t),

n

K(t),

0

of

where

i n the

be measurable and 1 Then t h e r e i s

cp E L

.

G(t)

a.e. on J.

y(t)

t h e unique point

K(t)

i s t h e b a l l with

and radius equal t o t h e distance

Rn

and appiy t h e known theorem t h a t t h e i n t e r -

section of two measurable functions i s a l s o measurable. Let

F: J xRn + cf(Rn) be such t h a t

(i)

f o r every fixed

n R , t h e function

x

E

t

E J,

F(t,x)

is

measurable on J; (ii)

f o r every fixed

upper semicontinuous on (iii)

and

qP2)

case

1_< p2

p1 = such t h a t

R”.

t h e r e exist functions

s = P1P2/(P1-

*,

THEOREM 4.15.2.

0

0

Let

F E

CY

p2 L , @ E Ls,

E

I P29 < PI’

< *,

For a given function x

t h e function

Iq <

03,

s

F(t,x)

is

where

1 I P1’P2 < co ( i n = p2 i s not excluded),

s a t i s f y assumptions ( i ) - ( i i i ) .

P1 Ln , l e t 292

Q(x)

denote t h e s e t of

4.15. GENERAL RESULTS FOR SET-VALUED MAPPINGS

y: J + Rn

a l l measurable f’unctions

y(t) E F(t,x(t))

llxll

Q(x)

(b)

Q(x) i s convex;

(C)

(a)

p2. Q(X) C Ln 9 Q(x) i s closed;

(e)

f o r every

5

k

on J.

x

We have t o show that f o r every

(a)

P1

a.e.

x + Q(x) defines a bounded mapping

Then t h e correspondence

Proof:

such that

E

P1 Ln

i s nonempty;

implies

t h e r e e x i s t s an M > 0

k > 0, lly[[

p2

C M

-

f o r every

y

E

such t h a t

Q(x).

We would prove only ( a ) , because (b) i s t r i v i a l l y true, ( e ) follows immediately from assumption ( i i i ) and obviously implies (c), and

kt

{xi] c

(a) w i l l

2

follow from Theorm 4.15.2 below.

be a sequence of measurable functions

admitting only a f i n i t e number of d i f f e r e n t values, convergent p1 La.e. on J. By assumption ( i ) t h e 11 (i = 1,2, ...) a r e measurable on J functions F ( t , x i ( t ) ) 1 and therefore, by Lemma 4.15.3, there a r e functions yi E Ln such t h a t yi($) E F ( t , x i ( t ) ) a.e. on J. For

t o a f’unction x

E

we have, by assumption ( i i i ) ,

Ilwi(t)l/

51

a.e.

on J.

Now

a straightforward application of Lemma 4.15.2 gives a sequence Gii(t) = C hikwk(t), k=i

293

i = 1,2

,...,

4. FUNCTIONAL ANALYTIC METHODS

which converges t o a function w E LL

n

a.e. on J.

It i s

e a s i l y seen t h a t t h e corresponding sequence

converges a.e.

t o a function

on J

Y(t> = w(t>[a(t> +

B(t>l!dt>llql

On the other hand, by assumption ( i i ) , f o r almost every fixed

t

and any

E J

>

E

F(t,xi(t))

there i s an integer

0

C

F(t,x(t) = [u

for

i

3 N(&,t).

+ kE

+ v: u

E

E

F(t,x(t)),llvll

F ( t , x ( t ) ) + kE,

and, by the convexity of yi(t>

2

E

F ( t , x ( t ) ) + kE,

THEOREM 4.15.3.

a.e. on J.

r

If

i n t o a Banach space

i

2 N(E,t)

i

2 N(E,t),

i s a l i n e a r transformation of then

E,

I'Q

It i s c l e a r t h a t f o r every

i s convex.

p1 {xi] C Ln , {z,]

imply

z

E

rQ(x).

E]

This proves t h e theorem.

maps

and i s upper semicontinuous.

TQ(x)

5

F(t,x(t)),

y(t) E F(t,x(t))

Proof:

such t h a t

Thus

yi(t>

so t h a t

N(E,t)

x

E

2

$

into

cf(E)

the set

Thus it remains t o prove t h a t f o r sequences

c E,

and x

E

p1 Ln , z

E

E,

For t h i s purpose, define 294

t h e conditions

zi =

mi,

yi E

a(.,)#

4.1 6 . SET-VALUED DIFFERENTIAL EQUATIONS

As i n t h e proof of Theorem 4.15.2

l o s s of generality, t h a t

[one can assume, without a.e. on 51 an

xi(t) + x ( t )

application of Lemma 4.15.2 y i e l d s a convergent sequence W

such t h a t

y(t)

E

F(t,x(t))

that is,

a.e. on J,

y

E

Q(x).

By assumption ( i i i ) , we have W

W

= a(t)

Denoting by

vi(t)

+

@(t)llx(t)llq, we f i n d e a s i l y t h a t applying Lemma 4.15.2,

k=i

4.16

-

-

+0 p2

+ 0.

and then, On t h e other

W

$ 1. = Z h i kIyi c m p l e te

llvi

(Iyi yII

we obtain

W

so that

v(t) = a(t) +

t h e l a s t sum and s e t t i n g

=

z = Iy and therefore

.

z

E

rQ(x).

The proof i s

SET-VALUED DIFFERENTIAL EQUATIONS Having developed necessary r e s u l t s concerning set-valued

mappings i n t h e preceding section, we a r e now ready t o employ t h e Kakutani-w Fan theorem t o prove an existence theorem f o r boundary value problems of set-valued d i f f e r e n t i a l equations. The r e s u l t s we present may be considered as extensions of c l a s s i c a l theorems on boundary value problems. Consider a system 295

4. FUNCTIONAL ANALYTIC METHODS

(4.16.1)

x'(t)

E

A(t)x(t) + F ( t , x ( t ) )

and a condition Lx = r.

(4.16.2) We shall assume t h a t

(a) A: J +A*, where A* i s the algebra of n x n matrices, which a r e measurable and integrable i n J ; +cf(Rn) such t h a t n (bl) f o r each x E R , the function F(t,x) i s measurable, n t h a t is, f o r each p E R , the distance from p t o the s e t (b)

F: J x R "

F ( t , x ) i s a measurable function (b2) f o r each t E J, the semicontinuous on R ~ , that i s , implies yo E F(t,xo); (b ) t h e r e exist function 3 integrable i n J such that

(a) (4.16.3)

E

in

i s upper

xi +xo, yi +yo, yi

E

F(t,xi)

a, f3 which a r e measurable and

R".

A, L,

and

x'(t)

r

are such that the problem

- A(t)x(t)

admits solutions for a l l Q~

F(t,x)

is a l i n e a r continuous operator from C = C[J,Rn]

(c) L i n t o R ~ ,r

i n J; function

f

=

f(t),

Lx = r,

belonging t o a l i n e a r manifold

L ;:

( e ) f o r every x E C, Q(x) i s the s e t of functions n y: J + R , measurable, such t h a t y ( t ) E F ( t , x ( t ) ) a.e. on J, and suppose t h a t Q(v) E cf(Qr), for each v E C w i t h Lv = r. Observe that condition ( a ) implies the existence of a which i s continuous and s a t i s f i e s unique function U: J X J +A*, the i n t e g r a l equation 296

4.16. SET-VALUED DIFFERENTIAL EQUATIONS

where

In i s t h e i d e n t i t y of

Also

i s a l i n e a r operator, continuous (and

U(.,s)

n R

compact), from

A*.

into

while t h e composition product

C,

% = L.Uf.,S) i s a l i n e a r operator from Rn

Furthermore, it i s known

m x n matrix has a generalized inverse and con-

t h a t every

%

sequently, t h e r e e x i s t matrices

For a fixed

:'I

f

+I'f

$ and hence can be

into

represented by an m x n matrix.

s

E

J,

of type

L1 n

= -U(t,s)q

into

such t h a t

l e t us define t h e l i n e a r operator

1

t

U(t,T)f(T)

dT

+

S

from

nxm

Lt

U(t,T)f(T) d7,

It i s easy t o prove by Ascoli's theorem

C.

t h a t t h i s i s a compact operator. Let us f i x a solution

c E Rn

+

Hr = u(t,s)(c

We claim t h a t i f

f E Or,

of

L$

= 0

and write

9).

t h e function

x=m+Hr i s a solution of problem (4.16.3).

t h e l i n e a r operator D(X

+

d/dt

Indeed, denoting by

- A(t)

from

t

Hr) = D [ - U ( t , s ) Q l S

297

c

into

D

Li we have

U(t,T)f(T) dT]

+

4. FUNCTIONAL ANALYTIC METHODS

+D

L t U(t,T)f(T)

+ D[U(t,s)(c + 9 1

dT

f t U(t,T)f(T) dT = f ( t ) .

= D

J S

Concerning t h e second equation i n (4.16.3), f E Or

means t h a t t h e r e e x i s t s an

7

E

we n o t i c e t h a t such that

Rn

n t

and hence

Js

+ L

U(t,T)f(T) d T +

= G [ r - L J

n t

U(t,T)f(T) dT1 + L

S

t

= r-L

r

U(t,T)f(T)

S

t

U(t,T)f(T) dT + L

J S

=

+

dT

r

t

U(t,T)f(T) dT

J S

r,

which proves our claim. By Theorem 4.15.2,

t h e correspondence x +Q(x) defines 1 a bounded transformation from C i n t o cf(Ln) and consequently, by Theorem 4.15.3,

r Q

and i s upper semicontinuous.

Hence

T: x + T ( x )

transforms

= I'Q(x)

+

C

cf(C)

[Hr}

i s a n upper semicontinuous transformation from C 29 8

into

into

cf(C).

4.16. SET-VALUED DIFFERENTIAL EQUATIONS

Let us denote by Vr, the linear manifold in C

vr

= [v E C: Lv = rl.

To prove the existence of solutions of the boundary value problem (4.16.1), (4.16.2), under hypotheses (a)- (e), it is enough to show the existence of a f’unction v E Vr such that v E T(v). By (e), T transforms Vr into cf(Vr). Indeed, the by (e); hence, transformation Q maps Vr into cf(Or), for each f E or, H + Hr is a solution of (4.16.3). Conl Y + Hr E Vr or transforms Or into vr- EH,~. sequent-, As r is linear, rQ(x) is convex for each v E Vr and since rQ is upper semicontinuous, it follows that rQ is closed for each v E Vr. This shows that T transforms Vr into cf(Vr). For x E C, z E T(x), we have by assumption (b ),

(4.16.4)

llzll

L llrll(ao + ~ ~ ) 1 1 ~ +1 1I I ~ r l l ,

where a. = JJ a ( s ) ds, Po,=’! JJ B ( s ) ds, and llrll of r. Let us suppose that Bollrll < 1 and set

K

P

= [X E

p =

C:

IIxII

_<

3

is the norm

P,

(~oIIrIt + IIH~II)(1-p0IIrII-’)

+ a,

where a > 0 is sufficiently large so that K fl Vr # #. P It follows easily from (4.16.4) that T(x) C K for P that is, T(KP) c KP, where T(KP) = T(x). each x E K P’ P Hence, we have T(KP n V,) c K I-7 Vr. AS K n vr is bounded, by Theorem 4.15.2, Q(K~ n v ,) P is bounded and as P is compact, it follows that T(K P fl V , ) is relatively compact in C. which Let us consider the convex closure of T(KD n V,),

uxEK

299

-

4. FUNCTIONAL ANALYTIC METHODS

we denote by co T(Kp

-

n V,);

then we readily obtain

n v,) C C O ( K ~n

c0 T(K,,

vr) =

K~ n

vr

as Vr, being the inverse image of the point r by the continuous operator L, is closed and K II Vr is closed and P convex. Hence T(=

T ( K ~n

vr))

c T ( K ~n

vr) = CO T ( K ~n v,).

Therefore the upper semicontinuous map T transforms the compact and convex set co T(K n Vr) into itself and conseP quently an application of Ky Fan fixed point Theorem 4.15.1 shows the existence of v E T ( K n Vr) such that v E Vr P We have thus proved the following theorem. and v E T(v).

4.16.1. Under hypotheses (a) - (e), problem (4.16.1), (4.16.2) has a solution if f3, = lJf3(s) ds is THEOREM

sufficiently small. If $ = m 5 n, problem (4.16.3) has a solution for each 1 f E Ln and each r and conversely. Let us call this hypothesis (f). If (f) holds, Qr = L1 n for each r E ? and in this case (a) is automatically satisfied. Hence from Theorem 4.16.1, we obtain the next corollary.

4.16.1. Under hy-potheses (a)- (c), (f), problem (4.16.1), (4.16.2) has a solution for ar!y r E ? if COROUARY

lJf3(s) d s

is sufficiently s m a l l . which .If $ = m = n, there exists only one matrix is the inverse of $ , for which the only solution of $C = 0 is c = 0 . Thus denoting by (h) the hypothesis that problem (4.16.3) has a unique solution for each f E Li and each n r E R , we have the following corollary. Po

=

COROUARY

Ltf

4.16.2. Uhder hypotheses (a)- (c), (h), problem 300

4.16. SET-VALUED DIFFERENTIAL EQUATIONS

(4.16.1),

so

(4.16.2) has a solution f o r each r = JJB(s) ds is sufficiently s m a l l .

Rn,

E

if

I n order t o i l l u s t r a t e the results, consider t h e following problem analogous t o the classical problem of Nicoletti,

(4.16.5)

x'

F(t,x),

E

xi(ti) = ri,

ti

E J,

Since the corresponding homogeneous problem x'

= 0,

xi(ti) = 0,

ri

E

R, i = l , 2

,...,n.

,...,n,

i = 1,2

has a unique solution x ( t ) a 0, an application of Corolhry

,...,

4.16.2 w i t h A(t) 5 0 and Lx = (x(tl),x(t2) x ( t n ) ) yields the existence of a t l e a s t one solution of (4.16.5) f o r arbitrary (rl,r2,. ,rn), provided F satisfies hypothesis (b) and

s,

..

= /,s(s>

-= 1.

ds

I n the case of the Cauchy problem

(4.16.6)

x'

E

F(t,x),

x ( t o ) = r,

to E J,

r

E

n

R

,

condition Po < 1 i s superfluous. It i s easy t o verify that replacing the usual norm i n .C by

we obtain, instead of (4.16.4) 11zo

f o r every

5 311x" + a() +

llrll

T(x). KLSO T ( K ~ )c K~ f o r P = 2(a0 + Ilrll). This implies t h a t i f F satisfies m o t h e s i s (b), then the Cauchy problem (4.16.6) has a t least one solution defined on J. Let us conclude with one more application. Consider the z

E

problem

(4.16.7)

X'

-A(t)x

E

F(t,x), 301

-

x(T) x(0) = 0,

4. FUNCTIONAL ANALYTIC METHODS

where

i s a periodic

A(t)

F: R x R n + cf(Rn)

n x n matrix of period

such that

F ( t +T,x) = F ( t , x ) .

T

and

Let hypotheses

( a ) , ( b ) hold and l e t

(4.16.8) with

f(t

X'

+

(4.16.3).

- A(t)x = f ( t ) ,

T) = f ( t ) ,

x(T)

- ~ ( 0 =) 0

be t h e problem s i m i l a r t o problem

Recall t h a t i f t h e homogeneous system has p e r i o d i c

s o l u t i o n s , then t h e a d j o i n t system has a l s o such s o l u t i o n s . Denoting by

Z

Assume t h a t

F ( t , v ( t ) ) E cf(OO) f o r each

t h e space of such s o l u t i o n s , we have

v E Vo,

where

Vo

i s t h e space of f'unctions from R i n t o Rn, continuous and p e r i o d i c with period T. Then by Theorem 4.16.1, Eq. (4.16.7) admits a t l e a s t one s o l u t i o n x ( t )

and t h e r e f o r e t h e set-valued

d i f f e r e n t i a l equation (4.16.7) has a l e a s t one p e r i o d i c s o l u t i o n of period 4.17

T.

NOTES AND COMMENTS Sections 4.1

- 4.3

a r e concerned with t h e r e s u l t s of Opial [2].

For r e l a t e d work, see Lasota and Opial [ 2 ] and Whyburn [l]. The contents of Section 4.4 are based on Conti [1,2]. further references,

see Conti's survey paper [21.

For

The results

contained i n Sections 4.5- 4.7 a r e adapted from Perov and Ribenko [2].

See Lepin and Nyshkis [l] f o r t h e r e s u l t s i n

Section 4.8.

The work i n Section 4.9 i s due t o Antosiewicz

[l].

Sections 4.10 and

and Yorke [ U I . [ 31.

4.11 c o n s i s t

of t h e results of Lasota

See, f o r more g e n e r a l results, Bernfeld e t al.

Section 4.12 contains t h e work of Schmitt [7 1.

Related

r e s u l t s on p e r i o d i c s o l u t i o n s may be found i n Schmitt [1,6,8] and Mawhin [1-31.

Sections 4.13 and 4.14 d e a l with t h e results

of Lasota and Opial [71 and Lasota [91. 302

For Lennna 4.14.1,

4.11. NOTES AND COMMENTS

see Plis [l]. For the results of Section 4.15, see Lasota and Opial [ j ] . The contents of Section 4.16 are taken from Grandolfi [ 11. For further results, see Lasota [51, Lasota and Olech [61, Ky Fan [11 and Chow and Lasota [l].

303

Chapter 5 EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

5.0

INTRODUCTION

We extend, in this chapter, some of the results of the preceding chapters, to boundary value problems associated with f'unctional differential equations. We begin by proving existence theorems in the s m a l l . We then extend the theory of differential inequalities and the modified function approach, to establish existence results in the large. A shooting type method is presented to derive existence and uniqueness results. We also consider the question of existence of solutions of quasilinear functional differential equations with nonhomogeneous linear boundary conditions. Finally, a more general problem is treated where the boundary conditions are prescribed in terms of bounded linear operators.

5.1 EXISTENCE

I N THE SMALL

are given functions that are bounded together with their derivatives on their respective domains. We shall Consider the BVP

( 5 1.1) (5.1.2)

x"(t) x(t)

=

=

cp(t)

f(t,x(t>,x(h(t,x(t)),x' on Ja,

x(t)

=

(t)), $(t)

We seek functions x where f E C[JxRxR,R]. satisfying the BVP (5.1.1), (5.1.2). Consider the Banach space B given by

3 04

on Jb> E

C(2)[J,R1

5 . l . EXISTENCE IN THE SMALL

with t h e norm I-

Here

I - \

BC"'[I,R]

C(l)[I,R] bounded on

means, as x u a l t h e class of f'unctions

such t h a t

implies that

y E BC(l)[I,R]

y,y'

are

I.

Recall from Section 1.1.1, t h a t if G(t,s)

is the Green's

function associated with the BVP

x"

x ( a ) = x(b) = 0,

= 0,

of t h e BVP

then the solution x ( t )

x ( a ) = x ( b ) = 0,

x" = p ( t ) ,

is of t h e form x ( t ) =

Iab

-

G(t,x) =

G(t,s)p(s) ds.

{

t

:i"yx)'

Also, l e t w ( t ) be defined on R

t

Define G(t,x) E

J,

J.

such t h a t

t t

-

b) ~ b-a Then c l e a r l y

w

(t-a )

p (

E

B,

Tx(t)

+

cp(a>, t

E

Ja,

E Jb, 6

J.

E B.

We next define t h e operator

x

as

=s,"

T

on B such t h a t f o r each

F ( t , s ) f ( s , x ( s ) , x ( h ( s , x ( s ) ) ) , x ' ( s ) ) ds

The following properties of

+ w(t).

T may be e a s i l y established as

i n Section 1.1.1:

305

5. EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

Tx(t)

(2)

i s twice continuously d i f f e r e n t i a b l e on 3 ;

(3) (W'W = f ( t J x ( t > i x ( h ( t , x ( t > ) J(~t )' ) on J; ( 4 ) T: B + B and the fixed points of T a r e solutions of t h e BVP (5.1.1) and (5.1.2); ( 5 ) T i s a continuous operator. Let R.

>0

M,N

be given such that

(w(t)l

(w'(t)( IN

5 Mj

We consider the closed, convex, bounded subset

Bo

on

of B

defined by Ix(t)l

Bo = [X E B:

_<

Ix'(t)l

_< 2N]

and we l e t q = [ s ~ P I ~ ( ~ , x , Y , t~ )EI J:J

If x

Irac(t>l

E BoJ

5

IYI 5

IzI

we can compute, a s i n Theorem 1.1.2,

((b-a)2/8)

t o conclude

1x1,

4

+ M J

5

I(m)'(t)l

5 m]*

t h e estimates

( ( b - a ) / 2 ) q +N,

-

(b a)*/8 5 M, We can then use Ascoli's theorem and

TBo c Bo;

provided t h a t

((b - a)/2) q 5 N. Schauder's fixed point theorem t o complete t h e proof of the following r e s u l t concerning t h e existence i n t h e small.

COROLLARY 5.1.1.

bounded on J x R 3

.

Assume t h a t

f E c[JXR3,R]

Then every BVP (5.1.1),

(5.1.2)

and i s has a

solution. The foregoing r e s u l t s remain v a l i d if we assume that

306

f

5.1, EXISTENCE IN THE SMALL

and x are vector-valued functions and I I denotes an appropriate vector norm. In fact, one can suitably state and prove a more general existence result when the derivative x' also contains a deviated argument like x'(g(t)). This we give as an exercise. EXERCISE 5.1.1.

State and prove an existence theorem for

the BVP x"(t> = f ( t , x ( t > y x ( h ( t , x ( t > > > , x (t>yx'(g(t>>), ' x(t) = cp(t),x'(t> = cp'(t> on Jay x(t> = Jr(t),x'(t) = Jr'(t) on Jb. Regarding the question of uniqueness of solutions, the following example illustrates the difference between the BVP

(5.1.1), (5.1.2) and its associated BVP without derivatives in the arguments Consider the BVP

.

x"(t)

-

=

x(t> 1 + IX(X(t>)l

for t < o

X = O

y

oet..

and

tLT.

It is easy to check that x(t) = c sin t is a solution of this problem for any c 5 0 whereas, the corresponding problem, nameXY, x"(t)

=

-

x/(l

+

[XI),

x(0)

=

x(7r)

0

5

t

< ny

= 0,

has only the trivial solution. Nonetheless, the uniqueness is guaranteed, if f satisfies an appropriate Lipschitz condition, as is common in these type of problems. A verification of this statement is left to the reader.

307

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

and f o r

t

E

Ix(t)l

(b

- a)/2]

,< kl,

<1

kl, ( h ( t , x )

- h(t,xl))

5 M I X - xl/.

(5.1.2) has a t most one solution x ( t )

Then t h e BVP (5.1.1), with

5

J, 1x1, Ixl(

5

Ix'(t)l

where

q = L1

k2

provided t h a t

2 q max[(b-a) /2,

+ L 2 ( l + k p ) + L3-

Let us now s i n g l e out f o r l a t e r use, a s p e c i a l case of t h e

BVP (5.l.l),

(5.1.2)

specifically, l e t and

Jb,

i n which f t

h(t,x)

- 7,

i s of retarded type. 7

> 0,

so t h a t

More

Ja = [ a - 7,aI

The corresponding BVP may be s t a t e d

a s i n g l e point.

as (5.1.3) (5.1.4) 5.2

x"(t) = f(t,x(t),x(t x ( t ) = cp(t),

t

E

- T),x'(t)),

[&-?,a],

x ( b ) = B.

EXISTENCE I N THE LARGE

Let us consider t h e second-order functional d i f f e r e n t i a l equation (5.1.1) together with t h e boundary conditions (5.1.2), with respect t o which, we shall define upper and lower solutions. DEFINITION 5.2.1.

A f'unction

second derivative on J (5.1.1),

(5.1.2),

CY E

B

having a continuous

w i l l be called a lower solution of

if

( i ) a " ( t >2 f(t,CY(t>,CY(h(t,CY(t>)),a'(t)), t E J, ( i i ) a ( t ) 5 cp(t) on Ja, a ( t ) 5 q ( t ) on Jb.

An upper solution @(t)

of (5.1.1),

(5.1.2) i s defined

s i m i l a r l y by reversing i n e q u a l i t i e s ( i ) and ( i i ) . We s h a l l e s t a b l i s h t h e following r e s u l t . THEOREN 5.2.1.

f

Consider t h e BVP (5.1.1),

s a t i s f i e s t h e conditions: 308

(5.1.2),

where

THE LARGE

5 . 2 . EXISTENCE IN

(i)

f(t,x,y,z)

(t,x, z 1;

is nonincreasing i n y

f o r each

3

( i t > If(t,x,y,z)I ,< 9 on J x R , ( i i i ) there exist constant functions

Q,

which a r e

f3

(5.1.2)

lower and upper solutions, respectively, of (5.1.1), such t h a t

Q

_< B.

Then there e x i s t s a solution x ( t )

(5.1.2) with a,< x ( t )

(5.1.1),

,< f3, t

of

R.

E

Let us define a modified f'unction F by

Proof:

If(t,ff,?,x')

x-a -

+

if

x

< a,

l + x

where

B Y

a Since

> B, if ~ ~ Y I B , i f y < a.

if Y

i s bounded, because of ( i i ) , by Corollary 5.1.1,

F

modified BVP x " ( t ) = F(t,x(t),x(h(t,x(t))),x'(t)), has a solution x ( t ) .

Also, by the definition of

the

(5.1.2) F and

assumption ( i i i ) , we have (5.201)

F(t,Q,a,O) _< 0

We now claim t h a t a f a l s e . Then, since

tl,t2

E

(a,b)

and x(t,)

=

5

,< x ( t ) _< B, t E Q 5 x ( t ) _< B, t

=

Suppose that t h i s i s

R. J,

x(t)

>

there e x i s t

tl < t < t2 x ( t 2 ) o r x ( t ) < a f o r tl < t < t2 and

such t h a t e i t h e r

B

F(t,B,P,O)*

f3

for

a = x ( t 2 ). We s h a l l deal with t h e f i r s t case, the arguments of the second case being similar. There e x i s t s a

x(t,)

=

tO,tl < to < t2, such t h a t

x(t) 309

-B

assumes a positive

5 . EXTENSlQNS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

maximum a t

to, with x ' ( t o )

Then because of (5.2.1),

= 0.

we o b t a i n

By condition (i)and t h e d e f i n i t i o n of

-

F,

we f i n d t h a t

2

x " ( t o ) 2 ( x ( t o ) @)/(1 + x ( t o ) )> 0. This c o n t r a d i c t s t h e f a c t t h a t x ( t ) @ assumes a p o s i t i v e maximum a t to. Therefore x ( t ) _< f3, t E R. Similarly, we conclude Q _< x ( t ) on R. This, however, implies t h a t

Hence

-

which assures t h a t (5.1.2).

x(t)

i s a c t u a l l y a s o l u t i o n of (5.1.1),

The proof i s thus complete.

The next r e s u l t i s concerned with t h e existence of s o l u t i o n s of t h e BVP (5.1.1),

(5.1.2) where a s s m p t i o n (ii)of Theorem

5.2.1 i s omitted.

THEOREM 5.2.2.

Assume f u r t h e r t h a t

( i )and (iii) of Theorem 5.2.1 hold.

satisfies Nagumo's condition on whenever

CY

5 x ( t ) _< @ , a5

[a,b]

x(h(t,x))

(5.1.2) has a s o l u t i o n such t h a t Proof:

and l e t

Consider t h e BVP (5.1.1), (5.1.2)

Q

5

relative to f3.

_< x ( t ) 5 B on R.

W e f i r s t d e f i n e t h e f u n c t i o n Fo by

where, as before,

310

a,

Then t h e BVP

f

f3

(5.1.1),

5 . 2 . EXISTENCE IN THE LARGE

'=I

-

Then

x

if

a-<xL@,

Q

if

x

y=t

-

< a,

Y a

t h e r e i s an

x(t)

such t h a t i f

N> 0

x " ( t ) = F,(t,x(t),x(h(t,x(t))),x'(t)) Q _< x ( h ( t , x ( t ) ) ) 5 B, then I x ' ( t ) l t h e function

Then F

F

Hence, by Theorem 1.4.1,

satisfies Nagumo's condition.

Fo

if Q I Y I B , i f y < a.

i s a s o l u t i o n of

and

Q

on

N

_< x ( t ) 5 B, J.

Now define

F by

a l s o s a t i s f i e s t h e Nagumo's condition and furthermore,

Also

i s bounded on JxR'.

F(t,a,cY,O)

5

0

5

F(t,@,B,O).

We thus conclude by Theorem 5.2.1 t h a t t h e f u n c t i o n a l d i f f e r e n t i a l equation

x " ( t > = F(t,x(t>,x(h(t,x(t>>>,x'( t ) ) together with t h e boundary conditions (5.1.2) has a s o l u t i o n

x(t)

satisfying

CY

Ix(t)

By t h e d e f i n i t i o n of

F,

_< B on R.

Hence a l s o

t h i s implies t h a t

a solution of (5.1.1), (5.1.2),

Ix'(t)l

5

N.

x ( t ) i s actually

completing t h e proof.

If t h e deviating argument i s independent of t h e s o l u t i o n

i t s e l f , that is,

h(t,x)

E

h(t)

may be assumed t o be functions of

only, upper and lower s o l u t i o n s

t

i n s t e a d of constants.

We

s h a l l merely s t a t e as exercises r e s u l t s corresponding t o t h a t situation.

In f a c t , we s h a l l r e s t r i c t ourselves t o t h e BVP

(5*1-3), (5*1*4). EXERCISE 5.2.1.

With respect t o t h e BVP (5.1.3),

l e t hypotheses (i)and (ii)of Theorem 5.2.1 hold.

311

(5.1.4),

Furthermore

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

suppose t h a t t h e r e e x i s t lower and upper s o l u t i o n s r e l a t i v e t o (5.1.3),

of (5.1.3),

Then t h e r e i s a s o l u t i o n x ( t ) a(t)

5

5

x(t)

p(t)

on

(5.1.4) s a t i s f y i n g

R.

(5.1.4),

Relative t o t h e BVP (5.1.3),

EXERCISE 5.2.2.

cu(t),@(t) on R.

a ( t ) _< p ( t )

(5.1.4) such t h a t

l e t a l l t h e hypotheses of Theorem 5.2.2 hold except t h a t

a,p

are not assumed t o be constant lower and upper s o l u t i o n s . there exists a solution x ( t ) a(t)

_< x ( t ) 5 p ( t ) on R.

5.3

SHOOTING METHOD

of (5.1.3),

Then

(5.1.4) such t h a t

Let us consider t h e second-order delay d i f f e r e n t i a l equation x"(t) = f(t,x(t),x(t- h(t))),

(5.3.1) where

where

x ( t ) = cp(t), cp

C[[-c,o],R]

E

5t5

T,

s u b j e c t t o t h e boundary conditions

h E C[[O,T],R+],

(5.3.2)

0

t

and

E

x(T) = A,

[-c,O],

- c = min[t

- h(t):

0

5t5

TI.

We

assume t h a t (1) f E C[[O,T] xR2,R],

f o r each

(t,x);

(2)

P,Po,q

-P(t)(xif

x

-

XI

C[[O,Tl,R],

E

- q(t)(Y-?)

sz, y < y

and

q(t)

5 f(t,X,Y) t

E

i s nonincreasing i n y

f(t,x,y)

2

0

such t h a t

- f(t,%?)5 - P , ( t ) ( x - 3 ,

[O,Tl;

(3) t h e s o l u t i o n u ( t )

of t h e i n i t i a l value problem

(m) (5.3.3)

u"(t) + P(t)u(t)

(5.3.4)

u(t)

i s p o s i t i v e on

5

0,

-c

+ q(t)u(t- h(t))

5t5

(0,TI.

312

0,

= 0,

U'(O+) = 1,

5.3. SHOOTING METHOD

LEMMA 5.3.1. Under assumptions (1)- (3), the rvP (5.3.1) with the initial conditions x(t) = cp(t), t E [-c,O], x'(O+)=s has a unique solution. The conclusion of Lemma 5.3.1 is immediate because hypothesis (2) implies that f satisfies a uniform Lipschitz condition with respect to x,y, which in thrn yields the existence and uniqueness of solutions of IvP's.

LEMMA 5.3.2. Under assumptions (1)-(3), the BVP (5.3.1), (5.3.2) has at most one solution.

Proof:

Let x(t), y(t) be two different solutions of (5.3.1), (5.3.2) and set m(t) = x(t)-y(t). By Lemma 5.3.1, Without loss of generality, we may therefore x ' (O+) f y' (O+). suppose that there exists a tl,O < tl _< T such that m(t) > 0, 0 on [-c,O]. 0 5 t 5 tl and m(0) = m(t ) = 0 . Also m(t) 1 Because of (2), we then arrive at the differential inequality

In view of assumption (3), it is possible to choose an r > sufficiently large so that ru(t) > m(t), 0 < t ,< tl, and since (5.3.3) is linear and homogeneous, ru(t) is again a solution of (5.3.3). Consider now the BVP

(5.3-5)

u"(t)

(5.3-6)

u(t)

+ p(t)u(t) p

0

+ q(t)u(t

on [-c,Ol,

- h(t)) u(t,)

0

= 0,

= 0.

Clearly m(t) is a laver solution and ru(t) is an upper solution of (5.3.5), (5.3.6). Recalling the fact that q(t) 2 0 , we conclude, on the basis of Exercise 5.2.2, that there is a solution y(t) of the BVP (5.3.5), (5.3.6) such that 313

5 . EXTENSION TO FUNCTIONAL DIFFERENTIAL EQUATIONS

However, y(t) =

y(t)

k(t)

> 0.

u(t,)

i s a solution of (5.3.3) and consequently

r" < 0 .

f o r some

This contradicts t h e f a c t t h a t

Hence the proof i s complete.

LEMMA 5.3.3.

Let

u(t), x(t,s),

solutions of t h e IVP's (5.3.3), on

[-c,o], x ~ ( o + )= s;

vl(O+)

= 1,

and

respectively.

v"

and

(5.3.4),

+

v(t)

(5.3.1);

be t h e x ( t ) = cp(t)

Po(t)v = 0 , v ( 0 ) = 0 ,

Then for

s" _>

s,

we have

The proof of t h i s lemma i s similar t o t h e proof of Lemma and hence we leave it t o t h e reader.

5.3.2,

LEMMA 5.3.4.

Lemma 5.3.3. Then

g(s)

Let

u(t), x(t,s),

M[u(T)

Let

-k

v(T)] = 2

and

v(t)

be a s i n

and

g(s) = s-M[x(T,s)-A].

has a unique fixed point. Let

s" >

g(g)

- g(s)

Proof:

Using (5.3.7),

s.

Then . . u

= s

- s - M[x(T,g) - x(T,s)].

we then see t h a t lg(s)

- !&-)I

_<

YlS

-4

y = (v(T) -u(T))/(u(T) + v(T)). Since 0 5 y < 1, g i s a contraction mapping and therefore has a unique fixed point

where S*.

In view of t h e preceding lemmas, i f we note t h a t

x(T,s*) =A,

we have proved t h e following r e s u l t . THEOREM 5.3.1.

Then t h e BVP (5.3.1),

Suppose t h a t hypotheses (1)-(3) hold. (5.3.2) has a unique solution which may

314

5.4. NONHOMOGENEOUS LINEAR BOUNDARY CONDITIONS

be computed by the shooting method. Assumption (2) above, may be weakened. by the following:

THEOREM 5.3.2. of Theorem 5.3.1.

Let us replace it

Assumptions (l), (3) - (5) imply the conclusion

Proof: We observe that in proving Lemma 5.3.2 and the first half of inequality (5.3.7), assumption (4) rather than (2) was employed. Thus the BVF (5.3.1), (5.3.2) has at most one solution and for 0

g>

s,

< (g- s)u(t) 5 x(t,g)

- x(t,s),

0

< t 5 T,

holds. Let s be fixed and let g -+w. Then, since u(T) > 0 , we obtain x(T,Z) -+ m. Keeping s fixed and letting s + -m, N

we conclude similarly that x(T,s) +-w. Assumption (5) implies that x(T, s ) A is a continuous function of s, which must cover the whole real line by the above argument. Hence there

-

exists an s*

such that x(T,s*)

5.4 NONHOMOGENEOUS LINEAR

=

A and the proof is complete.

BOUNDARY CONDITIONS

In this section, we shall be concerned with the question of existence of solutions of a quasilinear f'unctional differential equation subjected to nonhomogeneous linear boundary conditions. The equations considered include both functional differential equations of retarded type and of advanced type. 315

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

h,k _> 0

Let

and l e t

> 0.

T

As before, l e t

C[[O,T],Rn]

denote the Banach space w i t h the norm

llxllo = supO < t < T I l X ( t > l l ’ We need the following function spaces simultaneously and consequently, we s h a l l adopt the notation given below f o r convenience.

I n each of the above cases the norm i s the sup norm. x

E

C4

and

t

E

[O,T],

we define the element

xt

E

the relationship -h

xt(8) = x ( t +8), Let

and l e t

f E C[ [ 0, T I

.

x C3,Rn]

C

3

by

k.

n x n matrix with domain

be a continuous

A(t)

,< 8 5

For any

[O,T]

We consider the functional

d i f f e r e n t i a l equation x ’ ( t ) = A(t)x(t) + f(t,xt),

(5.4.1) Let

O(t),$(t)

domains

be continuous

[-h,O],

[T,T+k]

0

5t5

T.

n x n matrices with respective

such that

Q ( 0 ) = I = $(T),

where

I

i s the identity matrix.

l i n e a r operator on

C

into

Let

L be a continuous

We a r e interested i n the

Rn.

existence of a t l e a s t one solution x ( t ) of (5.4.1) which, n satisfies the following additional f o r a given r E R

,

conditions (5.4.2)

k ( t ) = r,

(5.4.3)

x ( t ) = cp(t)x(O),

0

316

5t5 -h

T,

5t5

0,

5 A. NONHOMOGENEOUS LINEAR BOUNDARY CONDITIONS

(5.4.4)

T

x ( t ) = $(t)x(T),

If

5t5

T

is a continuous matrix on

B(t)

IIB(t>rll,

Rn> and

we denote by

l i n e a r operator

+ k.

[a,b],

we l e t

llBl10 = S‘PtE[a,b]

t h e usual norm f o r a

((-((

L.

Recall t h e fact t h a t t h e solution of

x’

(5.4.5)

for every b

(5.4.6)

+ b(t),

= A(t)x

Lx(t) = r,

r

E

Rn,

i s given by

E C

Lt

x ( t ) = U(t)[L%(t)l-’[r-

+

u(t)

t

L(U(t)

u-l( s ) b ( s )

U-l(s)b(s) as]

ds,

and t h a t BVP (5.4.5) has a unique solution for every and every f E C if and only i f t h e BW

= A(t)x,

X’

XI

R”

Here U(t) is the fundamental

= A(t)x

such t h a t

Let us define a mapping way.

E

Lx(t) = 0

has only the t r i v i a l solution. matrix solution of

r

S: C4 + C 4

U(0) = I.

in the following

let

For each y E C4,

where (i) for each

t

E

x ( t ) is t h e unique solution

[O,T],

of t h e d i f f e r e n t i a l equation X’

= A(t)x + f ( t , y t ) ,

0

5 t 5 T,

s a t i s f y i n g t h e boundary condition

t ( i i i ) for t (ii) f o r

E [-h,O],

Lx(t) = r; x ( t ) = Q(t)x(O);

E [T,T+kl,

x ( t ) = Jr(t)x(T).

317

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

Noticing t h a t f o r each y

E C4, f ( t , y t ) i s a continuous function t for t E [,,TI, it i s clear, from the f a c t mentioned above, t h a t the mapping S i s well defined. We s h a l l next show t h a t the mapping S i s continuous on C4. Let y,z E C4. Then

of

Also, from (5.4.6) we obtain

Since U(t)

i s a fundamental matrix solution and

L

is a

continuous l i n e a r operator, there exist positive constants cy,

p,y, 6 such t h a t

-1

IlLll 5 6 -

Moreover, we have, llSY

- SZIll

5

Ilmll,llsY(o)

-< IlmIllIIsY and

318

-

- sz(0)Il

SZ/IO’

5.4. NONHOMOGENEOUS LINEAR BOUNDARY CONDITIONS

The foregoing three inequalities, then yield IISY

- SZII 5 maX[llQII1,

IIJrll,,ll@BT(wj

+

l)llf(t,Yt)

- f(t,zt)llO,

which, in view of the assumed continuity of f on [O,T] X C 3 shows that S is a continuous operator on C4. We shall impose conditions on f to ensure that for some

>

5 Bp, where B = [Y E C,+: P then it follows from (5.4.6) that p

0 , S(Bp)

I/Y(/~

_< PI. Let

y E Bp;

These inequalities imply that

where llO1lo

v1, and

q2

l/$l l.

are constants depending on a,p,y,6,T, Thus if p is such that

.

then S(Bp) 5 B y1 particular, p may whenever y E B P’ P be chosen this way provided f satisfies the condition that for sufficiently large p

(5.4.8)

Ilf(t,u)ll

5

(P-

IIuI~~

111)/11*,

whenever 5 p. We next verify that S ( B ) is sequentially compact. P Indeed, since S(Bp) 5 Bp, S(B ) is uniformly bounded. FurtherP more, if y E B then for t E [O,T] P’

319

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

and hence t h e assumption (5.4.8) on equicontinuous on

imply that

f

{Sy]

is

Moreover, as

[ 0, T I .

S y ( t ) = @(t)Sy(O),

for

-h

5t5

0,

and

it i s evident that and

[T,T

+ k].

{Sy} i s also equicontinuous on

[-h,O]

The sequential compactness i s therefore a

consequence of Ascoli's theorem. We summarize the foregoing considerations i n t h e following existence theorem. THEOREM 5.4.1.

Let

and s a t i s f y (5.4.8). solution f o r every

BVP x '

be continuous on

[O,T] xC3

Assume t h a t t h e BVP (5.4.5) has a unique n r E R and every bc C if and only i f the

= A(t)x, & ( t ) = 0 has only the t r i v i a l s o l u t i o n .

t h e BVp (5.4.1)

5.5

f(t,u)

- (5.4.4)

Then

has a t l e a s t one solution.

LINEAR PROBLEMS

I n t h e following section we shall continue our study of boundary value problems f o r f'unctional d i f f e r e n t i a l equations. Some preliminary notation w i l l be needed.

Let

Ch

denote t h e Banach space of continuous functions from into

Let

Rn,

L(t,cp)

where f o r

and

[a,b] xCh + R n

cp E Ch

f(t,cp)

be continuous mappings from

and f o r each

t

E

bounded l i n e a r operator from Ch function f ( t )

[ a - h,a]

defined on

[a,b],

320

[a,b], into let

let Rn. ft

L(t,cp)

be a

For any continuous denote t h e element

5 5 . LINEAR PROBLEMS

of

f t ( 8 ) = f ( t + 8 ) , 8 E [ a - h , a ] , t E [a,b].

defined by

Ch

We w i l l consider t h e two point BVP (5-5.1)

where

t

Y ' ( t > = L(t,Yt) + f(t,Yt),

and N

M

E

[a,bI

a r e bounded l i n e a r operators on

We s h a l l assume:

(%)

There e x i s t s a bounded integrable function

such t h a t

IIL(t,(p)ll

(H2) f(t,cp)

5

J(t)Il(p11,

Rn

E

[a,b]

and

J(t)

(p E

Ch.

[a,b] xCh

and s a t i s f i e s

u n SUPIJ(plJO-)m llf~t,P~ll/ll~l10 = 0, t

Ya =

t

for

maps closed bounded subsets of

i n t o bounded s e t s i n (5-5.3)

'h'

E

uniformly f o r

[a,b].

(H3 ) Solutions of t h e i n i t i a l value problem (5.5.1) with ch

e x i s t and a r e unique.

We w i l l make use of t h e properties of t h e l i n e a r equations

x'(t>

(5.5.4)

g

E

C[[a,b],Rn].

= L(t,xt),

For any i n i t i a l

q E Ch,

we write

x(q,g)(t)

as the solution of (5.5.5) s a t i s f y i n g xa(q,g) = q. For each and q E Ch a solution of t h e i n i t i a l value g E C[[a,b],Rn] problem (5.5.5) exists and i s unique.

The solution can be

represented as (5.5.6)

x ( q , g ) ( t ) = x(q,O)(t) + x(O,g)(t),

t

2

a - h*

From Gronwall's inequality, it follows, using (5.5.5) and (I-$), that

321

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

The following elementary result f o r t h e l i n e a r equation (5.5.4) gives a necessary and s u f f i c i e n t condition f o r t h e (5.5.2) t o have a s o l u t i o n .

BVP (5.5.5), LEMMA

5.5.1.

The two p o i n t BVP (5.5.5),

s o l u t i o n if and only i f denotes t h e range and

which maps

q

E

Ch

N%(O,g) X

E

(5.5.2) has a

JI + R(M+NX),

i s an operator on

i n t o t h e segment a t

t h e i n i t i a l value problem (5.5.4) with

Ch

R

where

defined by

of t h e s o l u t i o n of

b

xa

= q.

JI + R(M+NX). To show t h e BVP (5.5.5), (5.5.2) has a s o l u t i o n , it suffices t o e s t a b l i s h t h e e x i s t e n c e of a s o l u t i o n of t h e f u n c t i o n a l equation Proof:

(5.5.9) Using (5.5.6),

Suppose

%(O,g)

E

Mq + N%(q,g) =

E

'$9

'h'

we can w r i t e (5.5.9) as

(5*5-10)

Mq

+ NXq + N%(O,g)

=

'$a

N%(O,g) E Jr + R(M+NX). Conversely, i f t h e boundary value problem (5.5.5) and (5.5.2) This has a s o l u t i o n s i n c e

a s o l u t i o n x ( t ) on [ a - h,b], t h e n xa = q Consequently, N%(O,g) (5.5.8) and (5.5.9). completing t h e proof.

satisfies E

JI + R(M+NX)

We a l s o need t h e following r e s u l t on t h e operator LEMMA 5.5.2.

l i n e a r operator on

The operator Ch.

322

X

has

X.

i s a completely continuous

5.5. LINEAR PROBLEMS

Proof: For q1,q2

E

Ch and scalars

cy

and 0,

define

Immediately, we see, due to the linearity of L(t,(P), that z(t) satisfies (5.5.4) with za = 0. By uniqueness z(t) = 0, proving X is linear. Moreover, X is continuous due to the fact that solutions depend continuously on initial conditions. It remains to show X is compact. Let {R} be a bounded sequence in ch> 1l%llo 5 M. From (5.5.7)

Hence EX%]

is uniformly bounded.

Moreover

are equicontinuous and an application of the Hence EX%] Ascoli's theorem proves that X is completely continuous. Before stating our main results we need two lemma's which provide conditions on an operator so that the range of the operator is the whole space. We omit the proofs. LEMMA 5.5.3.

Let Y be a completely continuous mapping of a Banach space B into itself. If

(5*5*u)

ljJn SUP

llwll/lld < 1,

Il(Pll'm

then R(1- Y )

=

LEMMA 5.5.4.

B. Recall R(1-

Y)

means the range of I- I.

Let To be a contraction operator of a 323

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

Banach space

Let

into itself.

B

be a completely con-

T1

tinuous operator of a Banach space i n t o i t s e l f such t h a t

-

( 5 5 12)

~ ~ T l ( P ~ ~ / / k=P /0/.

Lim

l cp Il+

Then R(I

+

+ T1)

To

LEMMA 5.5.5.

= B.

Let

from a Banach space

B

T be a completely continuous operator into

Assume t h a t t h e only s o l u t i o n

B.

of t h e equation

(I + T)x = is x

z

E

Then f o r each

0.

y

E

t h e r e exists a unique

B,

such t h a t

B

(I + T)z Lemma 5.5.5

5.6

0

=

i s o f t e n r e f e r r e d t o as t h e F r e d h o h a l t e r n a t i v e .

NONLINEAR PROBLEMS We now consider a r e s u l t f o r t h e BVP (5.5.1),

THEOREM 5.6.1. Eq.

(5.5.1).

Assume hypotheses (H1)- ( H ) hold f o r

Let t h e operator

a r e given i n (5.5.2) and inverse.

(5.5.2).

(M

X i n (5.5.8),

Then t h e BVP (5.5.1),

3

+ NX),

where

M

and

N

have a continuous

(5.5.2) has a t l e a s t one

solution. Proof:

For each

q E Ch,

denote by

of t h e i n i t i a l value problem (5.5.1) with g(q)(t) = f(t,yt(q)).

Then y ( t , q )

y(t,q) y

a = q. Define i s a s o l u t i o n of t h e

nonhomogeneous l i n e a r equation (5.6.1)

x ' ( t > = L(t,xt) + g ( q ) ( t ) *

Thus 324

the solution

5.6. NONLINEAR PROBLEMS

A solution of the BVP i s determined by an i n i t i a l condition which is a solution of the functional equation

(506.3)

(M

+ NT)q

This may be rewritten as ( M

= JI,

JI

E Ch.

+ NX + N(T-

X ) ) q = Jr.

I? = (I4 + NX)'l which exists by hypothesis. can be rewritten as

Let

(5-6.4)

( I + l"(T-X))q = I'$,

Then (5.6.3)

$ E Ch.

We w i l l use Lemma 5.5.3 t o show the existence of solutions by

letting

Y = m(T- X)

and

B = C[O,h].

Since I T i s con-

tinuous, it i s sufficient t o show that T - X i s completely continuous i n order f o r Y t o be completely continuous. We now show T - X i s completely continuous. Let q E Ch and consider the solution y(t,q) of (5.5.1) w i t h ya = q. Then, by (5.5.3), (5-5.71, we have

where L = exp(/: R(s) as) and Mo is any constant such t h a t Ilf(t,cP)II 5 II9II0 + Mo f o r a l l t E [a,b]. An application of Gronwall's inequality then leads t o

Let

be

that from (5.5.7),

any bounded sequence i n Ch. Ubserve (5.6.2), and (5.6.5), we obtain

325

5 . EXTENSIONS TO FUNCTlONAL DIFFERENTIAL EQUATlONS

-

which shows

i s uniformly bounded.

{ (T X)R}

I n addition,

by (5.5.3)

Furthermore, f o r

t

E

[a,b],

we derive from (5.5.6),

(5.5.7)J and (5.6.5)

From t h i s estimate, (H1),

1%

( 0 J g (%)

and (5.6.5), we see, using (5.6.6),

) (lo

i s uniformly bounded. ~n a p p l i c a t i o n of A s c o l i ' s theorem y i e l d s the complete continuity of T - X . I n order t o apply Lemma 5.5.3, it i s s u f f i c i e n t t o show (5.5.11) holds. Suppose t h e r e e x i s t s a sequence of functions

that

s

E

Ch,

Ibt(g)II

l l ~ l +l

--1m

as

t h e r e e x i s t s an

~llcpll~for IIys

as n

n + m such t h a t

r >o

or

E

such t h a t i f

a l l t E [a,b].

(s) lo >r

o <

Let

+m.

<

pn

sup, < F J

llf(tJv)II

Since f o r each s E [ a , b l , 1l0 5 r J we a r r i v e a t

l/y, (s)

326

5

5.6. NONLINEAR PROBLEMS

Since E is arbitrary, (5.5.11) holds w i t h Y = T - X f o r any sequence cq, such t h a t the corresponding sequence B n as n + m . However, i f {@,I are bounded, then (5.6.7)

+m

implies

11%

lim

(O,@;(Qn)

= 0,

11% (lo

ll¶JO-)m

so again (5.5.U) holds.

)lo

This completes the proof of the

theorem.

5.6.1.

Assume that the two point BVP (5.5.4), (5.5.2) has only the t r i v i a l solution f o r Jr 2 0. If M has COROLLARY

a closed inverse, then the BVP (5.5.1),

Proof: -

(5.5.2) has a solution.

By t h e closed graph theorem, The relation (5.5.2) may be rewritten as

M-l

i s continuous.

Hence it is s u f f i c i e n t t o prove (I + M-hX)-’ e x i s t s i n order t o use meorem 5.6.1. To do t h i s we w i l l invoke k m a 5.5.5. Since X i s completely continuous, so i s M - h . Hence by hy-pothesis the only solution of ( I + M-lNX)ya = 0 i s ya s 0. follows.

Thus

(I + M - h )

321

has an inverse and the result

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

COROLIARY 5.6.2.

Suppose t h a t t h e mappings

- a;

that is

f ( t + ~ , c p ) = f(t,cp)

for a l l

(5.5.1) are p e r i o d i c with period L ( t +P,cp) = L(t,cp)

and a l l cp

and

L and f

P = b

in

t 2 a

If t h e only P-periodic s o l u t i o n of (5.5.4)

E Ch.

i s t h e i d e n t i c a l l y zero s o l u t i o n , then Eq. (5.5.1) has a P[a,m).

p e r i o d i c s o l u t i o n on Proof:

Let

M= apply Corollary 5.6.1.

I, N = -1,

Jr

and

=0

i n (5.5.2) and

The following e x e r c i s e provides t h e uniqueness o f s o l u t i o n s of t h e BVP by assuming a Lipschitz condition on

f(t,cp)

and

u t i l i z i n g t h e contraction mapping theorem. EXERCISE 5.6.1.

Ch.

E

f i n e d i n (5.5.8)

(5.5.2)

and

kt

assume (€$) holds.

f o r a l l cp

Consider t h e BVP (5.5.1),

If t h e operator

, has

M

-t

NX,

where

a bounded inverse and i f

s u f f i c i e n t l y small, t h e n the BVP (5.5.1),

(b

X

- a)

i s de-

is

(5.5.2) has a unique

solution. Hint: -

Consider t h e operator

Fq = W - r N ( T - X ) q j

r

where

(M+NX)-'

and

defines a contraction on condition on

f

defined by @ E

ch,

i s defined by (5.6.2). Show F by u t i l i z i n g t h e Lipschitz

T

5

and choosing

show a f i x e d p o i n t of

F

b

-a

s u f f i c i e n t l y small.

Then

F y i e l d s a s o l u t i o n t o t h e BVP (5.5.1),

(5.5.2). EXAMPLE 5.6.1.

We now e x h i b i t an example which i l l u s t r a t e s

328

5.6. NONLINEAR PROBLEMS

Theorem 5.6.1. (506.8)

Consider the functional d i f f e r e n t i a l equation

- 1 ) + f(t,y,),

y'(t) = Y ( t

t

[0,2]

E

subject t o the boundary conditions (5.6.9)

Yo =

1

ii Ye,

Y0'Y2

E cr-1,01.

Here L(t,yt) = y ( t - 1 ) , M = I , N = - I, a = 0, b = 2, h = 1, satisfying (H1) and JI = 0, and f E C[[O,2] xC[-l,O],Rn] (€$). For example, l e t f(t,cp) = A(t)kpl/i, 0 < < 1, Observe t h a t the i n i t i a l value problem for the A E C[O,21. homogeneous li ne a r equation X ' (t ) = x ( t

( 5.6.10)

- 1)

can be solved by the method of steps; that is, f o r any i n i t i a l function q E C[-1,0], we have (5.6.U)

x ( t ) = q(O) +

Jt

0

5t

- 1) as

du,

q(s -1) as,

Lt L ~ -- ~

q(s

+

1

1,< t

,< 2.

Thus Xq(@) is given by

xq(e) = L

~ + ~

and we h n ed i at e l y see t h a t

-

(I $X)

q(s

1)

s'plIqll

as au,

e

E [-1,01

IbqII < 4. Hence

-1 has a continuous inverse !&~d an application of

Theorem 5.6.1 yields the existence of a solution of the BVP (5.6.8), (5.6.9). EXERCISE 5.6.2.

Prove the existence of a periodic solution

of 329

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

where

0

<

(r

< 1.

Hint:

Apply Corollary 5.6.2 t o show t h e existence of a s o l u t i o n of period 2 ~ .

5.7

DEGENERATE CASES

I n t h i s s e c t i o n we consider (5.5.1) when t h e l i n e a r p a r t i s not a f u n c t i o n a l on

Ivbre s p e c i f i c a l l y , we consider

Ch.

t h e f u n c t i o n a l d i f f e r e n t i a l equations of t h e form (5 07.1) where

Y ' ( t > = A ( t ) Y ( t ) + f(tJYt)Y f

and A ( t )

i s a continuous mapping from is a n x n

[a,b] xCh i n t o Rn matrix function continuous i n t . We

s h a l l assume t h a t (H2) and

( 5 )hold.

We f i r s t consider t h e case when A ( t ) t h e operator

X

= 0.

I n t h i s case

reduces t o

(5.7.2)

xq = :(a),

i s the function i n

where

:(a)

q(a).

with t h e constant value

The following equivalent norm on q E Chy define

Ch

w i l l be u s e f u l .

For each (5.7.3)

IlqII, = Ilq(a)II + 11qll0.

For a l i n e a r operator

THEOREM 5.7.1.

T

If

on

r0

Ch

define

= (M+N)-'

IIPoNlla < 1, then t h e BVP (5.7.1), has a s o l u t i o n . and

330

e x i s t s , i s continuous, (5.5.2) with

A(t)

=0

5 .I. DEGENERATE CASES

Proof:

I n view of Theorem 5.6.1,

it i s enough t o show e x i s t s and is continuous where X i s given

(M+NX)’l

that

by (5.7.2).

Consider the m c t i o n a l equation

(5.7.4)

(M+W)q = CP, CP E Ch,

which may be rewritten a s

+

(M

N

+ N(X - 1))s = 9,

cp E Ch,

w h i c h is equivalent t o

(5.7.5)

I

+

rON(X-I) q =

r’p. 0

-

Hence, by Lemma 5.5.4, it suffices t o show t h a t roN(X I ) is a contraction, f o r then, the inverse of M + NX would e x i s t ,

For any

919%

Il(X

E

- I)ql-

ch (X

- I)Q211a =

-

-c

Ihl(a)

= Ils,(a)

lql

=

Since on

Ch.

~ ~ I ’ o N<~1 ~,a the operator Hence (5.7.5)

This implies

- I)q21/ - c12(a) II + Ils, - q21Io - q2 (a) I1 + llql - q2 1l0

11(X I)s,-

(X

- %JIa*

-

roN(X I) is a contraction

has a unique solution f o r every rp E Ch.

( I + roN(@- I))-1

exists.

The continuity

follows and thus (M+IlX)-l

is continuous. proof.

= (I

+

- I))-‘r0

TON(X

An application of Theorem 5.6.1 concludes the

EXAMPLE 5.7.1.

We show the existence of a solution of

the BVP

331

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

ma

y ' ( t ) = f(t,Yt), where

-

Mq(0) = q(0)

0 E [a-h,a]

:/

<(a), Nq(0) =

 1=

N = I

-

where

Let us assume t h a t b a h. Then killa < 1. Thus (M + N ) - l e x i s t s and s a t i s f i e s /1(M+N)-llla5 74 . k r e o v e r , llNlla < 4. 3 Hence II(M+N)-hlla _< II(M+N)-lllalIN/la < 1

Aq(8) =

q ( z ) dz.

and we m a y apply Theorem 5 . 7 . 1 t o conclude t h e existence of a solution of t h e BVP f o r any

(3).

s a t i s f y i n g (H2) and

f(t,yt)

We now consider (4.7.1) when A ( t ) # 0. which A ( t ) = 0, t h e assumption on ro r u l e s out t h e p o s s i b i l i t y For t h e case i n

This i s reasonable since the

of having periodic solutions.

equation

x'

0

has any constant as a solution with periodic Now we w i l l be i n t e r e s t e d i n establishing

boundary conditions.

Since we a r e now

t h e existence r e s u l t s of periodic solutions. looking a t t h e reduced equation

(5.7.6) t h e mapping X functions

t

x ' ( t ) = A(t)x(t), ql,%

[a,bl,

E

w i l l assign t h e same value t o t h e d i f f e r e n t E

as long as

Ch

Hence it i s

q,(a) = q2(a).

unreasonable t o expect t h e operator

M + NX t o be i n v e r t i b l e .

I n our previous r e s u l t we imposed t h e condition order t o insure the i n v e r t i b i l i t y of

M

+

Ilr0NI1

< 1 in

Here we present

NX.

a r e s u l t which w i l l allow f o r periodic boundary conditions. We now assume t h a t U(t)

and N

M

are

nxn

matrices and l e t

be t h e fundamental matrix of (5.7.6) with

i d e n t i t y matrix.

Define t h e operator

Xuq(0) = U ( b - a

THEOREM 5.7.2. Suppose t h a t f o r a l l

Let

+

0)q(e),

M

and N

0 E [a

i s nonsingular and t h a t

- h,a]

r(0)

I

332

5:Ch

U(a) = I,

+Ch

the

by

8 E [a-h,a].

be

nxn

t h e matrix

(M+NU(b - a

matrices.

( 'M +

+ 0))-'

NU(b

- a + 0))-'

satisfies

5 .I. DEGENERATE CASES

llrmuIlo =

(5.7.7)

Then the BVP (5.7.1)J

sup

a-hs91a

b(e)NU(b-a+9)11 < 1.

(5.5.2) has a t l e a s t one solution.

The s o l u t i o n y t ( s )

Proof:

of (5.7.1) can be represented

as

for

9 E

(5.7.9)

[a-hJa].

Let

To

TOq(@)= s ( a ) +

be a

mapping defined on Ch

rb-a+e

u-l(s)f

Ja

(s ,ys

by

(9)) as.

From (5.7.8),

(5.7.9), we have yb(q) 'XuT0q. Thus a solution of the BVP (5.7.1)J (5.5.2) can be found by solving the functiona l equation (5.7.10 ) Let

(M + JWuTo)q = JIJ

JI

E

C*

F be an operator defined by

(m)(') ClearlyJ

=

r(e)q(e)J

F i s invertible.

[a-hJal.

Therefore (5.7.10) i s equivalent

to

(5.7.11) Let

TO-IGT

(I + 1

rmu(To- 1 ) ) s

+T2

defined by

333

=

WJ

JI

E

Ch-

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

we w i l l use Lemma 5.5.4 i s a contraction and I'NX T u 2 is completely continuous and s a t i s f i e s (5.5.12). We again work

To o b t a i n a s o l u t i o n of (5.7.11), and show t h a t

FNXuTl

with t h e norm

ll*lla

introduced i n (5.7.3).

Then from (5.7.12), Il(T1q1

- TI921

one obtains, f o r any

(8)

II 5 -< =

Ilql(a) IlsJa)

- q2WII + - q2(a>I/ +

Let

q1,q2

E

Ch.

8 E [a-h,al

Ilsp) lPl

-

II

- q21Io

Ils,- q211a,

and thus it follows t h a t

IIT1ql

- T1% IIa 5 Ihl - q2 IIa-

This together with (5.7.7) implies Observe, from (5.7.13)

i s completely continuous.

I'NXUTl

and (H2),

Since

immediately have t h a t r%T2

i s a contraction.

t h a t t h e operator

%

i s continuous, we

T2

i s completely continuous.

From (5.7.9), we obtain

Using (5.5.4) and t h e Gronwall a - h s t s b I/U(t)II. i n e q u a l i t y we deduce t h e existence of p o s i t i v e constants

where and

K = sup

%

such t h a t

//yt(q)llo 5 Klllq/lo

3.

K2

f o r all t

It then follows t h a t

and hence with (5.7.7), we conclude t h a t

An a p p l i c a t i o n of Lemma 5.5.5 completes t h e proof.

334

K1

E

[a,b].

5 B. NOTES AND COMMENTS

EXERCISE 5.7.1. M

+ NU(b

- a+0)

Show t h a t t h e i n v e r t i b i l i t y of the matrix

f o r each

0 E [a

t h e following requirement: condition MX(a) each

z

+ NX(2)

( 5 -7.15) (H2),

can be accomplished by

Eq. (5.7.6)

= 0

subject t o t h e boundary

has only t h e t r i v i a l solution f o r

E [b-h,bl.

EXAMPLE 5.7.2.

where

- h,a]

Consider t h e one-dimens ional equation

Y’(t) = -Y@)

f ( t +.rr,yt) = f ( t , y t )

(?),

t

+ f(t,Yt), for a l l t

2

0

-’0,

and satisfies

together with t h e periodic boundary conditions Yo = Y$

(5.7.16)

Yo’Y,

Here M = 1, N = -1, U(t) = e (1-e -(n+e))-l

=

r(e>.

-t

E

c [-LO].

and thus

(M

+ NU(b - a+€)))-’

mreover,

Hence (5.7.7) holds and an inmediate application o f Theorem 5.7.2 y i e l d s the existence of a solution of the BVP (5.7.15))

(5.7 ~ 6 ) .

5.8

NOTES AND COIMEXCS

The material contained i n Sections 5.1 and 5.2 i s taken

from G r h and Schmitt [ l ] , while the work of Section 5.3 i s due t o De Nevers and Schmitt [l]. Section 5.4 consists o f t h e r e s u l t s of Schmitt [g].

The contents of Sections 5.5

are based on t h e work of Waltman and Wong [4].

- 5.7

For the tech-

niques of operator theory employed i n these sections, see Granas [l]. For r e l a t e d r e s u l t s concerning f’unctional d i f f e r e n t i a l equations, r e f e r t o Gustafson and Schmitt [ l ] , Fennel [2],

335

=

5 . EXTENSIONS TO FUNCTIONAL DIFFERENTIAL EQUATIONS

Fennel and W a l t m a n [ l ] ,Schmitt [ 4 ] , Norkin [ l ] J Mawhin [31J and H a l e [l].

336

Chapter 6 SELECTED TOPICS

6.0

INTRODUCTION

This chapter i s concerned with some s e l e c t e d t o p i c s of current i n t e r e s t .

F i r s t of a l l , we present i n a u n i f i e d

s e t t i n g Newton's method which i s of p a r t i c u l a r value i n applications.

We discuss as i l l u s t r a t i o n s two numerical techniques

of solving boundary value problems.

These a r e t h e Goodman-

Lance method and t h e method of q u a s i l i n e a r i z a t i o n , which have a t t r a c t e d considerable a t t e n t i o n i n recent years.

We then

consider nonlinear eigenvalue problems as an a p p l i c a t i o n of t h e angular function technique.

The n-point boundary value

problem i s investigated i n d e t a i l by studying n-parameter f a m i l i e s of solutions.

Under t h e assumption of uniqueness,

some r e s u l t s on existence of s o l u t i o n s a r e derived.

6.1 "TON'S METHOD Here we wish t o present an a b s t r a c t formulation of Newton's method in a u n i f i e d s e t t i n g and then i n subsequent sections, deduce, from t h e g e n e r a l r e s u l t s f o r Newton's method, as i l l u s t r a t i o n s , two numerical techniques of solving boundary value problems.

These a r e t h e Goodman-Lance method and t h e

method of q u a s i l i n e a r i z a t i o n , which have a t t r a c t e d considerable a t t e n t i o n i n recent years. Let

X

and Y

denote r e a l Banach spaces and suppose

t h a t t h e b a s i c equation t o be solved,

(6.1.1)

f ( x ) = 0,

i s given by a mapping of an open subset which i s continuously d i f f e r e n t i a b l e . 337

U

of

X

i n t o Y,

This means t h a t

f

has

6 . SELECTED TOPICS

a derivative

f'(x)

f o r every x E U

continuous mapping of

into

X

i s defined as t h e unique element i n ( l / h l l ) Ilf (x + h )

lim

L(X,Y)

- f (x) - f ' (x)h 11 =

(x)h I/: //h11

of f'(x)

f o r which

so t h a t , i n p a r t i c u l a r , Ilf'(x) I1 = sup ((If

L(X,Y)

A s usual,

Y.

is a

f'(.)

i n t o t h e Banach space

U

continuous l i n e a r mappings of

llhII-tO

such t h a t

5

11

0

.

It follows then from the mean value theorem t h a t t h e r e e x i s t s , f o r every

xo

with center

x

(If

(6.1.2)

E

u

and every

and radius

0

(x2) - f(xl)

E

>

r

0,

an open b a l l

B(xoJr) C U

such t h a t

- f ' (xo)(x2 - xl) 11 5

Elk2

- xlll

f o r any two points f

If

f"(x)

xlJx2 E B(xoJr). i s twice d i f f e r e n t i a b l e i n U,

of

f

a t any

U may be defined i n a n a t u r a l way with a continuous b i l i n e a r mapping of X x X i n t o Y, and consequently

IIf"

(X)

II =

SUP

[(If"

x

t h e second derivative

E

(X) (hlJh2)

II: I$,I

Furthermore, i f i n a d d i t i o n . llfl' (XI

5

(1 5 k,

1, IIh2 II

5

x

we deduce, as

E

U,

13

a consequence of t h e mean value theorem, t h a t

IIf'

(6.1.3)

f o r every open b a l l

(x2)

- f ' (x,) II 5 kIIx2 - xlII

B(xo,r) c U

and

XlJX2

E

B(X0").

Also,

as a consequence of Taylor's theorem, t h e r e r e s u l t s t h e inequality

(6.1.4) When

l/f(x2)-f(xl)-f'(x1)(x2-x1)II

5 3 kllx2-x111 2.

n m X = R , Y = R , our assumption implies that, each

p a r t i a l derivative continuous i n

U,

1s

Djfi, 1_< i 5 m, j5 n and f l ( x ) a t x = (x1,x 2 338

e x i s t s and i s

,...,xn )

is the

6.1. NEWTON'S METHOD

n

l i n e a r mapping of

R

J(x1,x2,

into R"

...,xn ) =

given by the Jacobian matrix D f (X j i 1 '

...,xn ).

m I n particular, when X = Y = R observe t h a t f ' ( x ) i s a l i n e a r homeomorphism of F? onto i t s e l f i f and only i f the Jacobian, det J(x1,x2,. .,x n ) i s different from zero. Let us now state a r e s u l t concerning the modified Newton's method.

.

THEOREM 6.1.1. Suppose t h a t f i s continuously d i f f e r entiable i n U and t h a t there is an xo E U f o r which f l ( x o ) i s a l i n e a r homeomorphism of X onto Y. men, f o r any a E (O,l), there i s an open b a l l B(xo,r) C U such that, i f

II 5 r ( l - a > ,

IIfl-l(xo)f(xo)

(6.1.5)

w .

there e x i s t s a unique x E B(xo,r) f o r which f ( 2 ) = 0. Moreover, f o r any sequence {Tn] of l i n e a r homeomorphisms of X onto Y satisfying

Ilf'

(6.1.6) f o r every n

(6.1.7)

,> 0,

-1 (x,)

f

' (xo) II 5 71

.. Tn-I (xn>J

n =

O j l j Z j a . . ,

{xn] of points i n B(xo,r) ll~-xn+lll ,< ( & / ( 3 - a ) ) III-xnll

(6.1.8)

f o r every n

-

t h e successive i t e r a t i o n s

xn+l = xn

define a sequence such t h a t to

1I IlT,

2 0. 339

which converges

6 . SELECTED TOPICS

Since f i s continuously differentiable i n U, there i s an open b a l l B(xo,r) C U f o r which x E B(xo,r) Proof:

implies

(6.1.10) We prove by induction that the i t e r a t i o n s (6.1.7) generate a sequence Ex,} i n B(xoJr). Indeed, from (6.1.6) we deduce that

(6.1.11) f o r every with

n > 0.

n = 0,

(6.1.12)

5 (3/(3 - Q')) Ilf'-l(x0) 11

(ITilll

Hence, i f

then

x 1 i s determined from (6.1.7)

Ilxl-xoll _< 111- Til(To - f f ( x o ) ) l l ~k'-l(xo)f(xo)ll

< 3 r ( l - a ) / ( 3- 0 ) by (6.1.5)~

(6.1.6) and (6.1.11), s o t h a t x1 E B(xo,r). Thus, suppose xlJ...,x are points i n B(xo,r) satisfying P Then (6.1.2) and (6.1.6) show ,p-1. (6.1.7) f o r n = 0,l

,...

that

(6.1.13)

IIf(xn+l)

II =

IIf(xn+l)

- f(xn) - Tn(~n+l- xn) II

-< ( 2 q 3 Ilf ' -l(Xo 1I1) holds f o r

llxn+l

- Xn I1

n = O,l,...,p-1.

Therefore, if x n=p,

i s determined from (6.1.7) w i t h P+l then (6.1.11) and (6.1.13) im-ply t h a t

(6.L14) for

llxn+2-xn+lll _< ( 2 4 3 - a ) ) l$+l-xnll

n = O,l,...,p-l.

Hence, i n particular,

340

6.1. NEWTON'S METHOD

and so, by (6.1.12),

xp+l

E B(xo,r).

This proves our assertion,

It follows t h a t the i n e q u a l i t i e s ( 6 . ~ 1 3 )(~6.1.14) hold f o r every

t o a point

and therefore

5

(6*1*16) f o r all n

2 0,

n

I I X ~ ~ - X ~ I (I( 3

-

-> 0

and

p

- ~ ' ) / 3 ( 1 - ~ ' ) ) ( & / ( -3a ) ) "

-> 1.

This implies t h a t

Ik1-x0II

{x }

converges n as a consequence

x E B(xo,r) f o r which f ( 2 ) = 0, of (6.1.13) and t h e continuity of f . If there were another

point

;i E B(xo,r)

f o r which

f(T) = 0,

we could deduce from

(6.1.2) and (6.1.10) t h a t

which i s absurd unless

x = x . Clearly, ( 6 . 1 . ~ ) and (6.1.16) N

-

imply t h a t (6.1.9) i s s a t i s f i e d f o r every

n

,> 0.

Since, by

construction,

( 6 . ~ 2 ) (6.1.6), ~ every

n

,> 0.

REMARK

and (6.1.n) show t h a t (6.1.8) holds f o r

The proof i s complete.

6.1.1.

If t h e open b a l l B(xo,r) C U

i s chosen as

i n t h e proof of Theorem 6.1.1, then (6.1.10) implies t h a t f ' ( x ) i s , f o r every x E B(xo,r), a l i n e a r homeomorphism of X onto Y. Thus, we rnw s e l e c t i n Theorem 6.1.1 t h e sequence {Tn] of l i n e a r homeomorphisms of

-> 0, Tn

X

and

Y

such t h a t , f o r every

Since (6.1.6) i s n then automatically s a t i s f i e d , the assertions of Theorem 6.1.1

n

= fl(z )

f o r some

remain v a l i d without change.

zn E B(xO,r).

I n p a r t i c u l a r , t h e proof of

Theorem 6.1.1 shows t h a t we m q always take every

n

0,

zn

=

i n which case the i t e r a t i o n s (6.1.7)

form

341

for assume t h e

6. SELECTED TOPICS

xn+l -- xn

(6.1.19)

-f

-1(xn)f(xn),

n = 0,1,.

.. .

These a r e t h e i t e r a t i o n s of Newton's method. I n Theorem 6.1.1,

REMARK 6.1.2.

choice f o r t h e sequence and

Y is t o let

the condition

of l i n e a r homeomorphisms of

{Tn]

Tn = To

a p a r t i c u l a r l y simple n 2 0.

f o r every

X

I n t h a t case

(6.1.6), which implies ~ / T ~ l * f l ( x o )5-uI /~( ~3 - u ) <$,

may be replaced by a weaker requirement t h a t homeomorphism of

onto

X

Y

-1

ITo

be a l i n e a r

To

f o r which

-1II < 1,

f'(xo)

and t h e conclusions of Theorem 6.1.1 may be strengthened as -1 follows: f o r any CY satisfying fl(xo) < c0 < 1, 0

there i s an open b a l l

/IT,

B (x , r ) 0

0

C

U

111

such t h a t , i f

lb0-1f(xo)lI < r ( l - a o ) , there e x i s t s a unique point and t h e i t e r a t i o n s

in

(6.1.7)

Bo(xo,r) f o r which generate a sequence Exn] E

N

-

f o r every

n

2

0.

of points

x such t h a t N

BO(xO,r) which converge t o

Ilx xn+lll

f ( g ) = 0,

N

CYo IIX

- xnll,

Observe t h a t , i n p a r t i c u l a r , we may take

= f ' ( x ) i n which case these assertions remain valid f o r 0 any CY E ( 0 , l ) . This i s t h e so-called modified Newton's 0

T

0

method.

342

6.1. NEWTON'S METHOD

COROLLARY 6.1.1. U

If

f

i s continuously differentiable i n

and, i n addition, there i s a constant

> 0 satisfying

k

( 6 . ~ 3 )then ~ the assertions of Theorem 6.1.1 hold r e l a t i v e t o

any open b a l l B(xOJr) contained i n U

f o r which

Mareover, under t h i s additional assumption, t h e convergence estimates ( 6 . ~ 8 )(6.1.9) ~ can be considerably improved f o r the i t e r a t i o n s (6.1.19) of Newton s method. EXERCISE 6.1.1.

in U

Suppose t h a t

and t h a t there i s a point

i s a linear homeomorphism of CY E

(o,+]

f o r every one point

X

f

x

0

i s twice differentiable

onto

E

I f there i s an

Y.

and a closed b a l l s ( x o J r ) c

x

f o r which f ' ( x o )

E U

u

such t h a t

E ( x O J r ) , then show t h a t there e x i s t s a t l e a s t

;; E E(xoJr)

f o r which

f(2) = 0.

Also show t h a t

the i t e r a t i o n s (6.1.19) are defined f o r every generate a sequence verges t o

x

Exn]

of points i n

such that, f o r every

n _> 0 and B(X J r ) which con-

-

n _> 0,

0

and

where the sequences

{an}, {rn }

343

are obtained by s e t t i n g

6 . SELECTED TOPICS

a, ro = r and for every n 2 0.

a. =

Hint: -

2 2 S ~ / ( ~ - ~ C Y ~ + ~rC U , =) ,a r n+l n n’

F i r s t show that i f the p i n t

x P’

f o r some p

2 0,

i s such t h a t the assertions a(xn,rn) C U, f (x,) is a l i n e a r homeomorphism of X onto Y, \If ‘-l(xn)f (x,) 11 5 r n ( l -an), (2/rn) an for every x E B(xn,rn) are IIf1-’(xn)II IP(xn)II true with n = p , then they remain t r u e with n = p + l f o r t h e point x determined from (6.1.19) with n = p . Then, since P+1 these assertions are t r u e f o r the point xo, use the induction argument t o complete the proof. 6.2

THE GOODM-LANCE METHOD

Newton’s method provides a convenient framework f o r deriving convergence c r i t e r i a f o r a variety of techniques f o r the numerical solution of two point boundary value problems of ordinary d i f f e r e n t i a l equations. As i l l u s t r a t i o n , we discuss briefly, i n t h i s section, the Goodman-Lance method and i n the next section, the method of quasilinearization. These two methods are t y p i c a l examples f o r t h e two groups into which a l l such numerical techniques w be divided. Though they are basically i t e z + i v e in the sense t h a t the solutions of the given problem appears as the limit of a sequence of solutions of auxiliary problems, they d i f f e r i n the way these auxiliary problems are chosen.

I n t h e Goodman-

Lance method, these latter problems are i t e r a t i v e l y generated i n i t i a l value problems f o r t h e same given d i f f e r e n t i a l equation, whereas in the method of quasilinearization, they are boundary value problems involving t h e same boundary conditions, f o r iteratively generated l i n e a r d i f f e r e n t i a l equations. I n order t o avoid umjmportant d e t a i l s , l e t us r e s t r i c t ourselves t o the problem of determining, i n a given compact 344

6.2. THE GOODMAN-LANCE METHOD

i n t e r v a l J = [O,T],

a solution

(6.2.1)

G

of t h e d i f f e r e n t i a l equation

f(t,x)

XI1 =

which s a t i s f i e s t h e given boundary conditions N

(6.2.2)

u ( 0 ) = a,

G(T) = b.

m m

We w i l l assume throughout t h a t and i s continuous on (6.2.3)

and f o r

JX?

(tJx,)J

1,

fx(t,x)

(tJx,)

E

exists

JXR~,

- J ( t , x 2 ) I/ 5 y b 1 - x211,

IIJ(t,xl)

where t h e Jacobian matrix a t

i s i d e n t i f i e d with

f E C[JxR ,R

(t,x)

J(t,x).

of

with respect t o

f

The norm

IIJ(t,x)II

t h e matrix norm induced by t h e given norm i n

x

is, as usual,

Rf".

The Goodman-Lance method i s based on t h e following construction.

Suppose t h a t

s o l u t i o n of (6.2.1) i n and

u,(T)

value

#

b.

unJ

f o r some i n t e g e r

> 0,

is a

f o r which

u ( 0 ) = a, uA(0) = xn n Then f i n d , f o r every i n t e g e r i E [l,m], t h e J

of t h e s o l u t i o n wi

wi(0)

n

in

J

of t h e a d j o i n t v a r i -

a t i o n a l equation (6.2.4) f o r which

= J*(tJun(t))x

XI'

and form t h e matrix W(un ) m i s an orthogonal basis of R If

w.(T) = 0, wj(T) = ei,

(wi(0)*ej),

1

where

ei

there i s a point

xn+l

(6.2.5)

w(un>(xn+l

such t h a t t h e s o l u t i o n ~'(0= ) xn+l

E

=

satisfying

R"

u

.

- xn> = un(T) - b

of (6.2.1) with

i s defined i n

J,

u ( 0 ) = a,

l e t t h a t s o l u t i o n be

u ~ + ~ .

Obviously, our aim i s t o generate i n t h i s way, from a given s o l u t i o n u 0 of (6.2.1) i n J with u 0 (0) = a , u0l ( 0 ) = "0, sequence {un} of s o l u t i o n s un o f (6.2.1) i n J with

345

a

6 . SELECTED TOPICS

u (0) = a, uA(0)

= x such that {un] converges uniformly in n n' J to a solution u" of (6.2.1) for which (6.2.2) is satisfied. We will show that this is indeed possible, under suitable assumptions. Let us first observe that, if (6.2.1) is a linear differential equation, say

(6.2.6)

X" =

A(t)x,

then any solution v of (6.2.6) in J integer i

E

[l,m],

to the solution w

equation

(6.2.7)

X" =

for which wi(T)

(6.2.8)

is related, for every

= 0,

wj(T)

wi(0)'v'(O)

=

i

of the adjoint linear

A*(t) x

=

ei by the equation

- ei-v(T),

wj(O).v(O)

-

as a consequence of the Green's formula. Hence, if (6.2.6)

has a solution u" in J such that u(0) = a, u'(0) = x" and ;(T) = b, and if u is any other solution of (6.2.6) for which u(0) = a, u'(0) = x, then in particular N

(6.2.9)

w. 1(O).(G-

for every integer i

[l,m];

E

x)

=

ei-(u(T)

- b) N

that is, x - x satisfies the

equation (6.2.10)

where Wo

W ( g - x ) = u(T)- b, 0

is the matrix (wi(0)*ej).

Thus, in this case,

the Goodman-Lance method does yield, in one step, the missing initial value x" = c ' ( 0 ) for the solution u" of (6.2.6) that satisfies the boundary conditions (6.2.2). In the general case, our assumptions imply, among others, the following facts. If (6.2.1) admits a solution uo with u0 (0) = a, u;)(O) = xo which is defined in J, then there 346

6.2. THE GOODMAN-LANCE METHOD

e x i s t s an open neighborhood

U

x E U,

u of (6.2.1) with

t h e unique s o l u t i o n

u'(0) = x

C

Rm

i s a l s o defined i n J.

xo such t h a t f o r every

of

Moreover, t h e r e l a t i o n which

a s s o c i a t e s with every

x

of (6.2.1) s a t i s f y i n g

u(0) = a, u ' ( 0 ) = x

d i f f e r e n t i a b l e mapping

E U

h

u(T)

t h e value

of

U

u ( 0 ) = a,

into

of t h e s o l u t i o n

i s a continuously

Rm.

We assert t h a t t h e Goodman-Lance i t e r a t i o n i s p r e c i s e l y t h e i t e r a t i o n of Newton's method f o r determining a point

x"

E

U

f o r which

(6.2.11)

h(x")-b = 0 .

Clearly, any such p o i n t

x

gives rise t o a s o l u t i o n

u ( 0 ) = a, c ' ( 0 ) = N

(6.2.1) with satisfies

N

x"

N

u

which i s defined i n

of J

and

Z ( T ) = b.

n 2 0 and i f un i s t h e corresponding s o l u t i o n of (6.2.1) i n J with u ( 0 ) = a, n u'(0) = x then t h e p a r t i a l d e r i v a t i v e D.h(xn) of h a t n n' J x f o r each j E [l,m], i s given by n' Indeed, i f

xn

E

_.

U

f o r some i n t e g e r

D.h(xn) = v.(T) J J

(6.2.12)

i s the solution i n J

x"

(6.2.13) f o r which each

= J(t,un(t))x

v,(O) = 0 , v l ( 0 ) = e,.

v.(T) J solution w

J

i

of t h e v a r i a t i o n a l equation

J

J

Since by t h e Green's formula,

i s r e l a t e d , f o r every i n t e g e r

i E [l,m],

t o the

of (6.2.4) by t h e equation

( 6.2.14)

e * v . ( T ) = -wi(0).e i

J

it follows t h a t t h e d e r i v a t i v e h ' ( x n )

j' of

h

a t xn

is

given by

( 6.2.15) where, as before,

h ' ( x n ) = -W(un) W(un)

i s t h e matrix (wi(0).e.). J

347

Thus,

6 . SELECTED TOPICS

h ' ( x ) i s nonsingular, the i t e r a t i o n of Newton's method n (6.1.19) i s

if

and t h i s i s equivalent t o t h e i t e r a t i o n (6.2.5) because h(xn) = un(T),

by definition.

Therefore, any of the r e s u l t s f o r Newton's method, such as Corollary 6.1.1 or Ekercise 6.1.1 w i l l yield automatically sufficient conditions f o r the convergence of the i t e r a t i o n s of the Goodman-Lance method.

We s h a l l formulate below only

a direction consequence of Corollary 6.1.1. THEOREM 6.2.1.

Suppose t h a t (6.2.1) has a solution uo

with uo(0) = a, u;)(O) = xo which i s defined i n J, U

C

x

Rm be an open neighborhood of

E U

the unique solution u

~ ' ( 0 =) x

of

U

e x i s t s i n J.

into

Rm

and l e t

holds f o r any x1,x2

E U.

0

h

-1

as the mapping x +u(T)

be any constant f o r which

If t h e Jacobian matrix

nonsingular and there a r e constants such t h a t

and l e t

such t h a t f o r every

of (6.2.1) with u ( 0 ) = a,

Define

B>

xo

CY

E

(0,l)

h'(xo)

and

is

r> 0

B(xo,r) c U, ~ ~ h ' - l ( x o ) h ( x o<) ~r(1~ CY), and

@I(h' (xo)((5 u/r, then there e x i s t s a unique point x" E B(xo,r) f o r which the solution u" of (6.2.1) i n J with u"(C) = a, G'(0) =

x"

s a t i s f i e s (6.2.2).

Moreover, the i t e r a t i o n s of

Goodman-Lance method a r e defined for every a sequence

{xn} of points i n

B(xo,r)

n 2 0 and generate which converges t o

x such t h a t t h e corresponding sequence {un 1 of solutions u of (6.2.1) i n J w i t h u ( 0 ) = a, uA(0) = x n n n converge t o w

u"

uniformly i n J,

and that

348

6.3. THE METHOD OF QUASILINEARIZATION

and N

Ilx- Xnll

_<

;

2n

1-CY

2"

for every n 2 0 .

Ilhl-'(xo)h(x0)ll

6.3 THE METHOD OF QUASILINEARIZATION Let us now discuss the method of quasilinearization as it applies to the problem of constructing a solution u" of (6.2.1) which is defined in J = [O,T] and satisfies (6.2.2). As before, we shall assume that f E C[JxRm,F?], fx(t,x) exists and is continuous on J xRm and (6.2.3) holds. Our aim, again, is to generate, from a given continuous mapping uo of J into R ~ ,a sequence {un] of continuous mappings of J into R" such that {u ] converges uniformly n in J to a mapping u" which is a solution of (6.2.1) in J satisfying (6.2.2). However, in contrast to the Goodmanfor every n 2 0, is not Lance method, the mapping u n+l' chosen as a solution in J of the nonlinear equation (6.2.1) but, instead, is to be a solution in J of the linear equation

(6.3.1)

X" =

J(t,u,(t))x

+ f(ttun(t))

- J(t,un(t))un(t)

for which

(6.3.2)

= a,

un+l(T)

=

b.

This requires, of course, that a continuous mapping uo of J into R" can be found such that not only the equation (6.3.1) with n = 0 but, moreover, each iteratively generated equation (6.3.1) with n 1. 1 admits a solution un+l which satisfies the boundary conditions (6.3.2). We will now show that this demand can indeed be met, under appropriate assumptions. We note first that a continuous mapping u" of J into F? is a solution of (6.2.1), (6.2.2) if and only if

349

6. SELECTED TOPICS

where G

i s the Green's function i n J X J

such that

(l/T)(T- t ) s ,

0

_< s _< t 5 T,

(l/T) t ( T - s ) ,

0

5t5s5

G(t,s) =

This may be forxmilated i n the following manner.

T.

Consider the

Banach space C = C[J,Rm] with the usual norm llull = suptd IIu(t)ll. Define cp as the mapping of C into it,self such that

(6.3.4)

cp(u): t + a

f o r every u

E

C,

+ -tT

(b-a)

-k

T

G(t,s)f(s,u(s)) ds

and s e t

o(u)

(6.3.5)

=

u - cp(u).

then any point u" E C i s a solution of (6.2.1) which s a t i s f i e s (6.2.2) i f and only i f

o(i-i)

(6.3.6) Our assumptions on

= 0.

imply that the mapping cp is continuously differentiable i n C. Indeed, from (6.1.2) we infer inrmediately that the derivative cp'(u) of cp a t u E C is the continuous linear mapping of C into i t s e l f such that, f o r every v E C,

t

+

-

f

LT

G(t,s)J(s,u(s))v(s) ds.

Hence the mapping 0 of C into i t s e l f , as defined by (6.3.5), is also continuously differentiable i n C. In fact, since evidently

350

6.3. THE METHOD OF QUASILINEARIZATION

for every u E C, where e is the identity mapping in C, we deduce f r m (6.2.3) and (6.3.7) that

(6.3.9) for any ul,u2 E C. Observe that, as a consequence of (6.3.7) and (6.3.8), @'(u) is, for some u E C, a linear homeomorphism of C onto itself if and only if @'(u)v = 0 implies v = 0 or, equivalently, if and only if the relation ( 6.3.10)

v(t)

T =-

G(t,s)J(s,u(s))v(s)

ds

for every t E J implies v(t) = 0 in J. Thus, Q'(u) is a linear homeomorphism of C onto itself precisely for those u E C for which the homogeneous linear differential equation

(6-3-u)

X" =

J(t,u(t))x

has no nontrivial solution v in J such that v(0) = 0 , v(T) = 0 or equivalently, for which the nonhomogeneous linear differential equation ( 6.3.12)

x"

=

J(t,u(t))x

+ w(t)

has, for every w E C, a unique solution v such that v(0) = a, v(T) = b. In particular, O'(u) is a linear homeomorphism of C onto itself for any point u E C for which IIcp'(u)ll < 1, and in that case IIO ' - l ( u )

II 5 ( 1 - (IT' (u>II I-?

We claim that, if the iterations of Newton's method for determining a point u" E C for which (6.3.6) holds, namely the iterations

(6.3013)

Un+l

6 . SELECTED TOPICS

are defined for every n > _0, then they are exactly the iterations of the method of quasilinearization. More precisely, the point u n+l E C obtained from (6.3.13) is then, for every n > 0, the unique continuous mapping of J into R" which is a solution of (6.3.1) satisfying (6-3*2)Clearly, if the iterations (6.3.13) are defined for every n l 0, then (0' -1(u,) exists for every n 2 0. Hence (6.3.13) implies by (6.3.5) that

(6.3.14) which, in turn, implies by (6.3.4) and (6.3.7) that

(6.3.15) un+l(t>

=

a

+

(t/T)(b

- a)

for every t E J. This shows that un+l is a solution of the linear differential equation (6.3.1) for which (6.3.2) holds. In fact, un+l is the only such solution, because the existence of (O'-l(un) assures, by the preceding remarks, that the homogeneous linear differential equation (6.3.11) corresponding to (6.3.1) has no nontrivial solution v in J such that v(0) = 0, v(T) = 0 . This proves our claim. Thus, sufficient conditions for the convergence of the iterations of the method of quasilinearization can be deduced directly from any of the general results for Newton's method, such as Corollary 6.1.1 or EXercise 6.1.1, by simply applying the latter to the mapping (0 as defined by (6.3.5). We state here the following result which is a direct consequence of Corollary 6.1.1. THEOREM

6.3.1. Let uo be the mapping of J into 352

F?

6.4. NONLINEAR EIGENVALUE PROBLEMS

for which u o ( t ) = a + (t/T)(b- a )

t

for every

E

J,

let

lk(t,uo(t))ll

be positive constants such t h a t

a,B

5

l l J ( t , u o ( t ) ) / / 5 B,

a,

t

E

J,

2 and suppose t h a t ( 1 / 8 ) ~ < ~ 1. If there exist positive constants

where

a

0

y> 0

<1

and

r

such t h a t

i s a constant for which (6.2.3) holds, then there

e x i s t s a unique continuous mapping

u"

of J

into

Rf"

with

Ilu"-uoll < r which i s a solution of (6.2.1) and (6.2.2) i n J. Moreover, t h e i t e r a t i o n s of t h e method of quasilinearization generate an i n f i n i t e sequence J

into

R"

{un]

which converges t o

u"

of continuous mappings of uniformly i n J

and Ilu- unll N

for every

6.4

a2n 0

1 5 aO

1-a:

such t h a t

8 aT2

1 - g1B T 2

n > 0.

N0NLINEA.R EIGENVALUE PROBLEMS

mploying t h e angular function technique, discussed i n Chapter 2, we s h a l l consider t h e boundary value problem PA

353

6. SELECTED TOPICS

3

2

2

where f,g E C[J XR ,R], J = [a,bl, A1 + A2 > 0 , and 2 2 B1 + B2 > 0 . We seek values of the parameter h for which PA possesses a nontrivial solution. Let us list the following assumptions. (H1) f,g E C[JxR 3,R], the solutions of (6.4.1),

(6.4.2) can be continued over J, and x(t) 0, y(t) the only solution of (6.4.1), (6.4.2) with x(to) = 0 ,

0

y(t 0 )

is

= 0 for any to E J. (H2) There exists a continuous f'unction H(t,u,v,h) 2 2 defined on JxDxR, where D = [(u,v): u + v = 11, such that

3 2 on J x R , where r

=

2

x

.

+ y2

We shall first prove an elementary result. !I'HEOm

6.4.1. Let (H1) and ($) hold. Suppose that l i m inf

(6.4.6)

A+

+m

Lb

G(t,h) dt

=

-m,

where G(t,h)

=

[SUP H(t,u,v,h):

Then there exists a sequence such that

Am <

{\I

<

(u,v)

DI.

of eigenvalues of PA

with l h k+m

354

f

\

=+O0,

6.4. NONLINEAR EIGENVALUE PROBLEMS

is nonnegative integer. If x(t,\), y(t,\) is an eigenflrnction pair f o r \, k 2 m, then x ( t , \ ) has a t least k zeros, while y ( t , \ ) has a t least k - 1 zeros on J. where m

Proof:

( x ( t ) , y ( t ) ) be a solution of (6.4.1) - (6.4.3). We set x = r cos 8, y = r s i n 8, w i t h 8 normalized so t h a t it s a t i s f i e s - ~ / 2< 8(a,h) 5 ~ / 2 . A simple calculation shows, i n view of (H2) and the definition of G(t,h), that Let

e’(t,h)

5

cos

H(t,

8,

s i n e,h) 5 G(t,h),

which, when integrated, yields

Ja

the condition

e(a,h) = arc tan(-A1/A2), (6.4.6) gives lim inf e(b,h) = -00.

Recalling t h a t

A+m

For each k = 0,1,2

4, =

[A>

,...,

define

0: e(a,h) = arc t a n

(:)--,

8(b,h) = a r c t a n

where B(t,h) is arc t a n y ( t ) / x ( t ) f o r the corresponding solution of (6.4.1)- (6.4.3). Since e@,h) is continuous i n h and limb, 0(b,h) = -m, t h e range o f 8(b,A) contains the interval

( -m,8(bto)).

Let m z

such t h a t Am i s nonempty.

The sets defining 8(b,\

\,

k

o

integer we may define

be the s e e s t

Then f o r k

2

m,

> m, are also nonempty, since

-1) > arc tan(-Bl/B2) - kT

h with 1 i m infh+m O(b,h)

=-m.

355

and 8(b,h) is continuous i n Finally, notice that e ( t , \ )

6 . SELECTED TOPICS

crosses each of the l i n e s a t l e a s t once t o give

0 = -(2j

x(t,\)

+ 1) ~ / 2 ,j

a t least

0(t,\)

crosses each of t h e l i n e s

so t h a t

y(t,A,)

has a t l e a s t

k

0 =

-1

k

- jT,

zeros.

= 0,1,.

zeros.

..,k-1,

Also,

j = 1,2

,...,k - 1 ,

The proof i s

therefore complete. EXERCISE 6.4.1.

Prove other p o s s i b i l i t i e s of Theorem

6.4.1, namely ( i ) replace (6.4.6) by lim infh-m =

Sab G(t,A)dt

( i i ) reverse t h e inequality i n (H2) and assume t h a t b G(t, A) d t = + m. lim SUP&+ -m,

I,

A drawback of Theorem 6.4.1 l i e s i n t h e function

For very simple systems,

G(t,A). f o r large

G(t,A)

may become constant

and consequently t h e condition (6.4.6) cannot

A

-

For example i n t h e case x ' = y, y ' = ( A + l ) x , 2 H(t,x,y,h) = - ( A + l ) x - y 2 so t h a t G ( t , A ) = - 1 f o r

be s a t i s f i e d . we have

large positive

A.

hood of

from t h e consideration t o get b e t t e r r e s u l t s .

x = 0

Clearly one needs t o eliminate a neighbor-

This i s exactly t h e point of t h e exceptional s e t i n the follow-

ing hypotheses. On t h e u n i t c i r c l e

D,

f i n i t e s e t of points, where angle

-T

< CY 5

For each

7T.

3

that

(i)

D

6

P u (aii= 1

E6 =

(H )

let

E = [a1, ...,a ] denote a P i s parametrized by t h e c e n t r a l

> 0,

6,Ui + 6).

There e x i s t s a f i n i t e s e t there i s a

K

t

E J,

(ii)

h if

2 0,

CY

E

E

E

of p i n t s on D

such

> 0 f o r which

H(t, cos a, s i n a , h ) for

let

6'

< -K

for a l l s u f f i c i e n t l y small 6,> 0,

G (A) = sup[H(t, cos a, s i n C Y , ~ )t: E J, a E D - E ~ ] , 6

356

6.4. NONLINEAR EIGENVALUE PROBLEMS

then

lim inf

h+m

G (A) = 6

f o r each s u f f i c i e n t l y small

-m,

6>0.

i s s a t i s f i e d , i f we take 5 )and K = 1. [T/2, -T/2], 6 <4 If (H1) - (H ) hold, then the THEOFSM 6.4.2. 3 In the example, (

E =

assertions

of Theorem 6.4.1 remain valid.

l i m infbm

We need only t o show t h a t

Proof:

B(b, h ) =

-00,

since the remainder of the proof of Theorem 6.4.1 applies. M, 8(b,A) > -2M, f o r be the exceptional s e t and define

Suppose the contrary, t h a t i s , f o r some h> 0.

E = [a1,.

Let

F = [ql

,...,cpP 3 =

..,aP ]

1< i

[ai-2rj:

< p,

j = 0,1,..., M I .

< K(b - a)/kp(M+ 1 ) . For each h 2

l e t ti, i = 1 p, denote the unique time such t h a t B(t*i' h ) = cp.1 6. Notice t h a t ti,tr depend on h and are well defined because

Choose

6

,...,

8'

< -K when -

8 = 'pi 2

B(t,h)

-

In f a c t ,

6.

el(t,h) < -K,

so t h a t

0,

t

E

[ti,t21,

t

on

1. - K ( t y -

ti)

i s s t r i c t l y decreasing i n

[ti,"].

We have also

-

-26 = B(t?c,h) 1 B(ti,h) and, as a r e s u l t , we conclude t h a t (b

- a)/2(M+ 1)p.

i n t e r v a l s i n J,

i

Since there are a t most

5

26/K <

p(M+ 1) such

the measure of t h e s e t

T (h) = [t: t 1

satisfies

t*- ti

E

J,B(t,h)

m(Tl(h)) < ( b - a ) / 2 ,

E6]

so the measure of

T2(A) = [ t : t E J, B(t,h) 351

E

Esl

6 . SELECTED TOPICS

- a)/2.

i s g r e a t e r than

(b

so t h a t by ($),

it follows t h a t

5

e(bJh)

Since m(T2(h))

G&h)

=

- mJ

However,

- m ( T l (+h Gg(h)m(T2(h))n ))

>

(b-a)/2

f o r every

h and lim infh+,

h > 0 f o r which 8(b,h) < -2MT.

there e x i s t s a

This contradiction proves the theorem.

6.5

n-PARAMETER FAMILIES AND INTERPOLATION PROBLEM3 In Section 2.1, it was shown f o r second-order s c a l a r

equations t h a t the uniqueness of t h e two-point BVP implies i t s

An analogous theorem e x i s t s f o r the third-order

existence.

equation; however, the question of uniqueness implying existence n

remains open f o r the n-point BVP,

2 4.

I n t h i s section, we

prove under t h e additional assumption of " l o c a l s o l v a b i l i t y , t h a t the uniqueness does i n f a c t imply t h e existence f o r the n - p i n t BVP.

Our main technique i s the theory of a n-parameter

f a m i l y of functions.

We s t a t e our r e s u l t f o r t h e f i r s t order n-dimensional system; a s p e c i a l case of t h i s , of course, i s t h e nth order s c a l a r equation.

THEOREM 6.5.1. (6.5.1)

x' = f(t,x), x (t.) = c 3' O J

(6.5.2)

t 1 < t2 < (A)

Consider the boundary value problem

-.-< tnJ where f(t,x)

E

.

j = 1,. . J n J

x = (xo,xl,...,x

n n C[(a,b)xR J R 1;

358

n-1

).

Assume

6.5. n-PARAMETER FAMILIES AND INTERPOLATION PROBLEMS

a l l s o l u t i o n s of (6.5.1) e x i s t on

(B)

(a,b);

t h e r e e x i s t s a t most one s o l u t i o n of (6.5.1) and

(C) (6.5.2) f o r a l l

< tn;

c

E

3

and t . J

R

t h e BW (6.5.1),

(D)

every point

(6.5.2)

to E (a,b);

(a,b)

E

a r e l o c a l l y solvable a t

t h i s i s , f o r every

e x i s t s an open i n t e r v a l

tl < t2 <

where

to E (a,b)

I ( t O ) , t O E I ( t O ) E (a,b)

there

such t h a t

(6.5.1) and (6.5.2) has a s o l u t i o n f o r a l l d i s t i n c t and a l l

(6.5.2) has a unique solution.

Then every BVP (6.5.1),

We s h a l l need some d e f i n i t i o n s and lemmas f i r s t . DEFTNITION 6.5.1.

A set

F

of functions

f E C[(a,b),R]

i s s a i d t o be an n-parameter family on (a,b) i f f o r every s e t of n d i s t i n c t points tl < t2 < * . * < tn of (a,b) and every s e t of element

n

fo E F

numbers

cl,.

satisfying

.., cn

f ( t ) = c

(6.5.3)

0

DEFINITION 6.5.2.

j = l,...,n.

3’

3

t h e r e i s one and only one

A family

F of functions

f E C[(a,b),R]

i s s a i d t o be a l o c a l n-parameter f a m i l y i f f o r every

to E (a,b)

t h e r e i s an open i n t e r v a l ,

such t h a t

t h e s e t of r e s t r i c t i o n s

f E F

an n-parameter f a m i l y on

I ( t O ) , t O E I ( t O ) C (a,b) f l I ( t o ) of t h e elements of

is

I(tO).

Observe t h a t t h e conditions s o l u t i o n s of t h e BVP (6.5.1),

(B) and (D) imply t h a t t h e

(6.5.2) form a l o c a l n-parameter

I n order t o obtain t h e r e s u l t of Theorem 6.5.1 it i s s u f f i c i e n t t o show t h a t F i s indeed an n-parameter f a m i l y

family on

F.

(a,b).

Our next r e s u l t shows t h i s i s indeed so.

Thus

Theorem 6.5.1 i s incorporated i n t h e following more general setting.

359

6 . SELECTED TOPICS

THEOREM 6.5.2.

functions on

(a,b)

F = { f } be a f a m i l y of continuous

satisfying

i s a l o c a l n - p a r a m e t e r f a m i l y on

(i)

F

(ii)

if

has a t most Then

Let

f,g

a r e d i s t i n c t elements o f

c - 1 zeros on

and

F, then f - g

(a,b).

i s an n-parameter f a m i l y on

F

(a,b)

(a,b).

To prove Theorem 6.5.2 we s h a l l need some lemmas. LEMMA 6.5.1. f -g

(i)

If

has

n-1

F be an n-parameter family on

Let

f,g

are d i s t i n c t elements of

zeros on

(a,b),

each o f i t s zeros. (ii)

fo(t) = fo(t,tl

if

element of

(Cl,

then

2 n + 1 variables f o r n Cn) f R

.

...,

t

such t h a t

changes signs at

f -g

,...,t n ,cl ,...,cn )

determined by (6.5.3),

F

function of i t s

< tn < b,

then

F

fo

i s t h e unique i s a continuous

(a,b), a

E

(a,b).

< tl <

9 . 9

We s h a l l p r o v i d e p a r t o f t h e proof i n t h e following exercise. EXERCISE 6.5.1. Hints:

Prove Lemma 6.5.1.

For p a r t ( i ) assume

f

-g

does not change s i g n

a t some x 1' L e t ( a ' , b t ) be an i n t e r v a l having x1 i n i t s i n t e r i o r and containing none o f t h e abcissas of t h e o t h e r n-2 f(xt)

p o i n t s of i n t e r s e c t i o n .

#

g(xf).

Let

restricted t o n

-2

Show t h a t i f

f

the other

Ft

( a t , b t ) of t h e members of

(at,bt)

such t h a t

F

passing through

p o i n t s o f i n t e r s e c t i o n and through

(xt,g(xf)).

i n t e r s e c t s any element

(at,b')

e x a c t l y one time, then

f-g x2,x3

Then choose

f 2 E F'

and

g E Ft

changes sign.

by t a k i n g two p o i n t s fl

9

Choose x t

be t h e one parameter f a m i l y o f functions

such t h a t such t h a t

360

a'

in

This can be done

< x2 < x1 < x < b t fl(x2) = g ( x , )

3

and

.

6.5. n-PARAMETER FAMILIES AND INTERPOLATION PROBLEMS

-g

Show t h a t i f f f (x ) = g(x3). 2 3 then f l = f 2 at some point on

does not change signs, (%,?). But t h i s i s impossible

since f l and f2 belong t o F’. For ( i i ) show that f o r any sequence

) converging t o

“lk’ ‘ ’s9 ) (t’T4tJ* * J z nk’ slk’’ - ’snk

,...,

{T%,T~

,...,Tn ,

,...J ~ n )thenk converges t o f(t,zl, ...,zn’

( T ~

T~~

k

s1

sn) uniformly f o r t E (a,b)}. This can be shown by using t h e r e s u l t s of the first p a r t of t h e 1-a. S1>*..Y

TXMMA 6.5.2.

Aasume the conditions of Theorem 6.5.2 and

let

Then

Sl

is an open subset of R2n ,

-.-

Proof: Let toE (a,b) and sl < s2 < < sn be points of the open i n t e r v a l I ( t O ) . Recall f / I ( t O ) forms an nparameter family and l e t 7

f(t) = f(t,

be t h e unique element of For fixed to and sl, defined by

S1’

6

,...>C

n)

F satisfying

...,sn

We f i r s t show t h a t

..., n , c 1

rp

f ( s ) = cj, j = l,...,n, d consider the maming rp: A x R n + a

i s a continuous fbnction.

...

..,

sufficient to show t h a t f (t, sl, ,sn, cl,. continuous function of ( t , cl,. ,c ). kt J n i n t e r v a l contained i n (aJb) such that ti is

..

361

It i s

cn ) i s a be any closed contained in

6 . SELECTED TOPICS

the i n t e r i o r of

and to E J.

J

an open i n t e r v a l I ( x ) an n-parameter f a m i l y . ~

Ui=,

such t h a t

,...,k

E

restricted t o

F

f(t, s

I(xi) U I ( t O ) . Observe t h a t

...,

i s a continuous function of

J,

define

I(x)

form

some k.

On

rl,

E

I(%)

I(tO)n

,...,xn , c 1,...,"n)

on I ( t O )XR" (t, cl, cn I ( t O ) intersects I(%) f o r apply @ m a 6.5.1 by considering any points I(%) having the values that i s ,

because of Lemma 6.5.1.

...,rn

x

There e x i s t s a f i n i t e number of x2, I k such t h a t Ui=, I(xi) covers J. Let S =

i = 1

k

For each point

Now

3' ... ,sn, c1 ,...,cn), j 1,...,n. A p p l y now Lemma J 6.5.1 t o f ( t , r1 ,...,rn, al ,...,an) f ( t , s1 ,...,sn, c 1,...,cn ) on I(%). Hence f ( t , s1 ,...,sn, c1 ,...,cn) i s continuous f o r ( t , cl, ...,cn) on I ( t O ) U I(%). We may continue t h i s process a.

= f ( rj,

sl,

=

=

a f i n i t e number of times and obtain the continuity of

,...,

f ( t , s1 sn, cl,..",cn) on S x R n . Hence we see t h a t cp i s continuous on A X R ~ . We immediately have t h a t cp i s one-to-one on AxR" because of condition ( i i ) i n Theorem 6.5.2. Since A x R n i s open i n R2n then cp(A xRn) = i s open i n R2n by the Brouwer invariance of domain theorem. We now generalize 6.5.1 as follows. LEMMA

6.5.3.

Let

F be a family of continuous functions

on an open i n t e r v a l

f(t)

(i) (ii)

the s e t

has a t most

n

such t h a t

i s open and

Cl C R2n

f,g

if

(a,b)

are d i s t i n c t elements of

-1

zeros on

F,

then

f -g

(a,b).

%en

(a) If f , g (a,b),

then

f

(b) If

element of

F

-g

E

F

and. f - g

has exactly n - 1

zeros on

changes signs at each of i t s zeros.

f ( t ) = f ( t , tl satisfying

,...,tn, c1 ,...,cn)

f ( t i ) = ci, 362

then

f

i s the unique i s a continuous

6.5. n-PARAMETER FAMILIES AND INTERPOLATION PROBLEMS

2 n + l variables for t

function of i t s

(tl,

f

-g

...

..

,tn, C1'.

Proof: -

'Cn)

Let

vanishing at

there e x i s t s

neighborhood about o r contained i n of

51 we

an h

E

rnw

(a,b)

and

52.

E

f,g

E

F be d i s t i n c t elements of

t

=

tl,

...,tn-l

> 0 such t h a t

k

E

tk. Choose any s

(a,tk)

pick an

E

i+l. Suppose

i n some deleted (tk-l,tk)

if

k

>1

k = 1. Because of t h e openness

if 6

ti c t

where

-g >0

f

F with

> 0 so

s m a l l such t h a t there exists

F having t h e property t h a t

for

h ( t j ) = f ( t j )= g ( t j ) h(s) = f ( s )

h(t

and

-

-

k

+&)

j =

#

k

g(tk+E).

Observe that f h and g h each vanish a t n - 1 points. However, t h e continuity of f,g, and h imply t h a t either

o r g - h vanishes on (s, tk + & ) since f - g > 0 i n some deleted neighborhood of tk (which we may assume contains mis contradicts ( i i ) tk+ E ) The proof of the second part i s the same as i n Exercise f -h

.

.

6.5.1. U M M A 6.5.4.

Let

F be a s e t of continuous f'unctions

Then F is an n-parameter f ( t ) on an open i n t e r v a l (a,b). family on (a,b) i f it has properties ( i ) and ( i i ) of Lemma 6.5.3, t h e property (iii)

(6.5.4)

...

i s any sequence of

fi(t)

5 fi+l(t)

for i = 1,2,...

fi(t)

2 fi+l(t)

for

if

on a compact s e t

fl,f2,

[Q',p] c (a,b),

363

i = 1,2,...,

then e i t h e r

F satisfying

or

6 . SELECTED TOPICS

exists on

f ( t ) = lim f i ( t )

(6.5.5)

(a,b)

and

f

or

lim I f i ( t ) l =

(6.5.6)

on a dense set of

m

i-tm

(a,b)

and the property

to E (a,b) i s not bounded above o r below. (iv) for a l l

Proof:

To show

F

f i c i e n t t o show t h a t

Q .

..,t n )

the s e t

S ( t O ) = { f ( t o ) : f E F}

i s an n-parameter f a m i l y it i s sufn = A XR Let k, where 15 k _< n,

.

..,

..,

(cl,. c ~ - ~ c ~, + ~ , .cn ) be fixed and l e t S = ( Ck: (tl,. .,tn, cl,. .,c n ) E Q ) . We m e d i a t e l y have t h a t S i s open from (i)of Lemma 6 5 - 3 *

(tl,.

E

and

A,

.

S

i s a l s o closed.

{c, }

satisfying

We now show t h a t consider a sequence

c

k

E

j

{c

sequence and without loss of generality

ck

3

k

j -tm.

BY ~enrma6.5.3,

j.

m

ck

as

j

If

We may assume

and

4c

.

#

k, f j ( t k ) = c k

if 9

f

E

j

F

k j

S for j

5

= 1,2,.

..

i s a monotone

}

for a l l

c

kj+l and s a t i s f i e s f ( t ) = cm for j

j

then

m

j

r=O,

where

i s not empty

S

...,k - 2,

a = to and b = tn+l. Thus by condition ( i i i ) e i t h e r

(6.5.5) holds, and ck E S o r (6.5.6) holds. Assume (6.5.6) holds. By ( i v ) we can f i n d some f satisfying

f (t,)

>

ck

>

.

f (t,)

J

3 64

for a l l j

.

E

F

For s u f f i c i e n t l y

6 5 . n-PARAMETER FAMILIES AND INTERPOLATION PROBLEMS

hrge

we have i n view of (6.5.6) and (6.5.7)

j,

that

f-f

3

vanishes on either s i d e of tk a r b i t r a r i l y near tk and also arbitrarily near tm f o r rn f k. Thus, f o r large j, f - f j has n zeros on (a,b), a contradiction t o ( i i ) of Lemma 6.5.3. Thus S i s closed. Hence S = (-m,-) inplying n(a,b) = A(a,b) xRn. The proof of Theorem 6.5.2 thus follows from &mmas 6.5.1- 6.5.4. For our further discussion we shall consider the Bvp

x(n) = (6.509)

...

JX(

f(t,X,X',

x ( t i ) = ci,

The results of Theorem 6.5.1,

d ) ,

,...,n.

i = 1

applied t o the BVP (6.5.8) and

(6.5.9), remain true without the local solvability condition ( i i i ) f o r n = 3 (as w e l l as n = 2). Thus, we have the following theorem.

THEOREM 6.5.3.

Consider the BVP

Assume t h a t ( i ) f(t,x1,x2,x3)

E

C[(a,b) x R 3. , R l

( i i ) solutions of i n i t i a l values of xm = f(t,x,x',x") extend on (a,b) (iii) there exists a t most one solution of the BVP (6.5.10) f o r each ti' i =1 J 2 J 3 J where tl < t2 < t 3 and

each

c

i

E

R.

Then there exists exactly one solution of t h e BVP (6.5.10).

365

6 . SELECTED TOPICS

As we have pointed out, we have from Theorem 6.5.1 that (A)- (D) imply the existence of solutions of BVP (6.5.8) and (6.5.9). Under assumptions (A)- (C) we have seen that the following condition is equivalent to (D): is a sequence of solutions of (6.5.8) (D1) If {x,(t)}L=l which is uniformly bounded and montone on a compact interval [c,d] C (a,b), then limit x (t) = x(t) is a solution of k (6.5.8) on [c,d]. Another condition under assumptions (A) - (C) to (D) and (D1) is

equivalent

If {%(t)};=, is a sequence of solutions of (6.5.9) which is uniformly bounded on a compact interval [c,d] C (sib), (i1(t)} then there is a subsequence {xk (t)} such that {x, (D2)

j

converges uniformly for each 0 _< i 5 n-1. For

j

it can be proved that if the BVP (6.5.8), (6.5.9) satisfies (A)- (C), then it also satisfies (D2) (from which the existence of solutions follows). For completeness of discussion we now state a result for the k-point BVP, (6.5.8) and

(6.5.ll)

n

= 2,3

i x (t.) J

=

-1, l < j < k ,

cij, k

6.5.4. If solutions of initial value problems for (6.5.8) are unique and if the n-point BVP (6.5.8), (6.5.9) THEOREM

have unique solutions which extend throughout (a,b), then the k-point BVP (6.5.8), (6.5.11) have unique solutions. Thus in view of our previous results, conditions (A)- (C), uniqueness of initial value problems f o r (6.5.9) and either 366

6.6. NOTES AND COMMENTS

(D), (D,), or (D,) imply the existence and uniqueness of npoint BVP's and k-point BVP's.

6.6

NOTES AND COMMENTS

The contents of Sections 6.1- 6.3 are taken from htosiewicz [41. For a detailed treatment of quasilinearization techniques, see Bellman and Kalaba [l]. Section 6.4 consists of the work of Macki and Waltman [l]. For further results using similar methods, see Macki and Waltman [ 21 and Hartman [ 51. The results of Section 6.5 concerning n-parameter families are adapted from Hartman [41. In particular, Lemma 6.5.1 is due to Tornheim [l]. See also Klaasen [ 3 ] and Jackson [TI for related results. The results relating the k-point and n-point problems may be found in Hartman [l], Jackson and Klaasen [31 and Jackson The special results on the third-order equation can be found in Jackson and Schrader [4,5]. See also Klaasen [2].

[a].

361

BIBLIOGRAPHY

AKO, K.

[l]

Subfunctions f o r ordinary d i f f e r e n t i a l equations V, Funkcial. W a c . 12(1969), 239-249.

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383

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INDEX A A-matrix, 226 Adjoint equation, 346 A priori estimates, 248 Angular function technique, 116

B Boundary value problems (see also Existence of solutions), boundary conditions of, linear--2point,2,10,81,95,101 linear-n point, 208, 358 general linear, 200 nonlinear, 34,36,113 sets for, 159 integral conditions for, 221 Bounds on derivatives, 26,73, 76 Brouwer invariance of domain theorem, 362

C Carathdodory theorem, 205 Contingent equations (see also Set-valued differential equations), 176 upper and lower solutions of, 182 Nagumo condition for, 185 existence theorems for, 186

in the large, 31 over infinite interval, 44 with general linear boundary conditions, 10 with nonlinear boundary conditions, 36 Existence of solutions of second order vector equations, in the small, 71 in the large, 81,83,85 Existence of solutions for first order twodimensional systems, estimation of number of solutions for, 121 finite, 136 infinite, 142 Eigenvalue problems, 353

F Functional differential equations, existence results for, 304, 308, 320, 324,332 uniqueness results for, 328 Funnel of solutions, 94, 153 b

Generalized spaces, 225 Schauder fixed point theorem, 226 contraction mapping theorem, 227 Goodman-Lance method, 344 Green’s function, 3, 246

D

H

Degree theory, 270 Dependence of solutions on boundary data, 186

Hausdorff metric, 176 Helly’s theorem, 238 Homogeneous function, 120 Homotopic, 263,271

E Egress points, 157,178 Existence of solutions of second order scalar equations, of two point boundary value problems, in the small. 2

I Interpolation problem, 208,358 Invariant sets, 177

385

INDEX

P

K Kakutani-Ky Fan fixed point theorem, 289 Kneser’s theorem, 95,188

Periodic solutions of functional differential equations, 328 of ordinary differential equations, 269 of set-valued differential equations, 301 Perron’s method, 62 Perturbed linear problems, 213

L LeraySchauder theory, 263 L p solutions, 234,243 Lyapunov functions, for existence theorems, 39, 87,265 for bounds on derivatives, 41,79 for boundary conditions, 145

Q Quasilinearization method, 349 Quasimonotone property, 72,209

M

S

Minimal and maximal solutions, 122 Modified functions, scalar-valued, 18 vector-valued, 69,89

Schauder fixed point theorem, 207 Set-valued mappings, 278,289 Set-valued differential equations, 176, 283,295 Shooting method, 95,312 Subspaces of solutions, 256 Super- and subfunction, 47 properties of, 52 relation with lower and upper solutions, 47

N N-parameter families of functions, 359 Nagumo’s condition, for scalar equations, 25 for systems, 74,76 for contingent equantions, 185 to determine boundary conditions, 166 Nicoletti problem (see also Interpolation problem), existence for, 209,211 uniqueness for, 2 11 Nonhomogeneous systems, of ordinary differential equations, 200,25 6 of set-valued differential equations, 283 of functional differential equations, 316,321 Nonlinear boundary conditions (see Boundary conditions)

T Topological methods, 154, 160 Tonelli’s theorem, 2 36

U Underfunction, 6 3 Uniqueness, 15,127 Uniqueness implies existence, 94, 102, 106,109,358 Upper and lower solutions, 12 Upper semicontinuous mappings, 278,292

W

0

Wazewski topological method, 157, 168, 180

Over function, 6 3

38 6

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